*553*
*94*
*3MB*

*English*
*Pages 159*
*Year 2015*

*Table of contents : Preface......Page 13Acknowledgments......Page 154.1 Curves and Surfaces in Rn Given by ODEs......Page 174.2 Volume of Geodesic Balls......Page 214.3 Holomorphic Geometry......Page 254.4 Kähler Geometry......Page 295.1 Basic Properties of de Rham Cohomology......Page 335.2 Clifford Algebras......Page 425.3 The Hodge Decomposition Theorem......Page 475.4 Characteristic Classes......Page 576 Lie Groups......Page 61Basic Concepts......Page 62Lie Algebras......Page 63The Exponential Function of a Matrix Group......Page 71The Classical Groups......Page 76Representations of a Compact Lie Group......Page 78Bi-invariant pseudo-Riemannian Metrics......Page 85The Killing Form......Page 87The Classical Groups in Low Dimensions......Page 92The Cohomology of Compact Lie Groups......Page 97The Cohomology of the Unitary Group......Page 100Smooth Structures on Coset Spaces......Page 103The Isometry Group......Page 107The Lie Derivative and Killing Vector Fields......Page 111Homogeneous Pseudo-Riemannian Manifolds......Page 116Local Symmetric Spaces......Page 118The Global Geometry of Symmetric Spaces......Page 1208.1 Homological Algebra......Page 1278.2 Simplicial Cohomology......Page 1378.3 Singular Cohomology......Page 1408.4 Sheaf Cohomology......Page 144Bibliography......Page 149Authors' Biographies......Page 155Index......Page 157*

SSyntheSiS yntheSiS yntheSiS L LectureS ectureS ectureS on on on M atheMaticS atheMaticS and and and S StatiSticS tatiSticS tatiSticS MatheMaticS Series Series SeriesEditor: Editor: Editor:Steven Steven StevenG. G. G.Krantz, Krantz, Krantz,Washington Washington WashingtonUniversity, University, University,St. St. St.Louis Louis Louis

Aspects Aspects of of Differential Differential Geometry Geometry II II Peter Peter PeterGilkey, Gilkey, Gilkey,University University Universityof ofofOregon, Oregon, Oregon,Eugene, Eugene, Eugene,OR OR OR

JeongHyeong JeongHyeong JeongHyeongPark, Park, Park,Sungkyunkwan Sungkyunkwan SungkyunkwanUniversity, University, University,Suwon, Suwon, Suwon,Korea Korea Koreaand and and Institute Institute Institutefor for forAdvanced Advanced AdvancedStudy, Study, Study,Seoul, Seoul, Seoul,Korea Korea Korea

GILKEY •• PARK PARK •• VÁZQUEZ-LORENZO VÁZQUEZ-LORENZO GILKEY

Series Series SeriesISSN: ISSN: ISSN:1938-1743 1938-1743 1938-1743

Ramón Ramón RamónVázquez-Lorenzo, Vázquez-Lorenzo, Vázquez-Lorenzo,University University Universityof ofofSantiago Santiago Santiagode de deCompostela, Compostela, Compostela,Santiago Santiago Santiagode de deCompostela, Compostela, Compostela,Spain Spain Spain

Book Book BookII II IIdeals deals dealswith with withmore more moreadvanced advanced advancedmaterial material materialthan than thanBook Book BookIIIand and andisisisaimed aimed aimedat at atthe the thegraduate graduate graduatelevel. level. level.Chapter Chapter Chapter 444deals deals dealswith with withadditional additional additionaltopics topics topicsin in inRiemannian Riemannian Riemanniangeometry. geometry. geometry.Properties Properties Propertiesof of ofreal real realanalytic analytic analyticcurves curves curvesgiven given givenby by byaaa single single singleODE ODE ODEand and andof of ofsurfaces surfaces surfacesgiven given givenby by byaaapair pair pairof of ofODEs ODEs ODEsare are arestudied, studied, studied,and and andthe the thevolume volume volumeof of ofgeodesic geodesic geodesicballs balls ballsisisis treated. treated. treated.An An Anintroduction introduction introductionto to toboth both bothholomorphic holomorphic holomorphicand and andKähler Kähler Kählergeometry geometry geometryisisisgiven. given. given.In In InChapter Chapter Chapter5, 5, 5,the the thebasic basic basic properties properties propertiesof of ofde de deRham Rham Rhamcohomology cohomology cohomologyare are arediscussed, discussed, discussed,the the theHodge Hodge HodgeDecomposition Decomposition DecompositionTheorem, Theorem, Theorem,Poincaré Poincaré Poincaré duality, duality, duality,and and andthe the theKünneth Künneth Künnethformula formula formulaare are areproved, proved, proved,and and andaaabrief brief briefintroduction introduction introductionto to tothe the thetheory theory theoryof of ofcharacteristic characteristic characteristic classes classes classesisisisgiven. given. given.In In InChapter Chapter Chapter6, 6, 6,Lie Lie Liegroups groups groupsand and andLie Lie Liealgebras algebras algebrasare are aredealt dealt dealtwith. with. with.The The Theexponential exponential exponentialmap, map, map,the the the classical classical classicalgroups, groups, groups,and and andgeodesics geodesics geodesicsin in inthe the thecontext context contextof of ofaaabi-invariant bi-invariant bi-invariantmetric metric metricare are arediscussed. discussed. discussed.The The Thede de deRham Rham Rham cohomology cohomology cohomologyof of ofcompact compact compactLie Lie Liegroups groups groupsand and andthe the thePeter–Weyl Peter–Weyl Peter–WeylTheorem Theorem Theoremare are aretreated. treated. treated.In In InChapter Chapter Chapter7, 7, 7,material material material concerning concerning concerninghomogeneous homogeneous homogeneousspaces spaces spacesand and andsymmetric symmetric symmetricspaces spaces spacesisisispresented. presented. presented.Book Book BookII II IIconcludes concludes concludesin in inChapter Chapter Chapter888 where where wherethe the therelationship relationship relationshipbetween between betweensimplicial simplicial simplicialcohomology, cohomology, cohomology,singular singular singularcohomology, cohomology, cohomology,sheaf sheaf sheafcohomology, cohomology, cohomology,and and and de de deRham Rham Rhamcohomology cohomology cohomologyisisisestablished. established. established.

ASPECTS OF OF DIFFERENTIAL DIFFERENTIAL GEOMETRY GEOMETRY II II ASPECTS

Differential Differential DifferentialGeometry Geometry Geometryisisisaaawide wide widefield. field. field.We We Wehave have havechosen chosen chosento to toconcentrate concentrate concentrateupon upon uponcertain certain certainaspects aspects aspectsthat that thatare are are appropriate appropriate appropriatefor for foran an anintroduction introduction introductionto to tothe the thesubject; subject; subject;we we wehave have havenot not notattempted attempted attemptedan an anencyclopedic encyclopedic encyclopedictreatment. treatment. treatment.

We We Wehave have havegiven given givensome some somedifferent different differentproofs proofs proofsthan than thanthose those thosethat that thatare are areclassically classically classicallygiven given givenand and andthere there thereisisissome some somenew new new material material materialin in inthese these thesevolumes. volumes. volumes.For For Forexample, example, example,the the thetreatment treatment treatmentof of ofthe the thetotal total totalcurvature curvature curvatureand and andlength length lengthof of ofcurves curves curves given given givenby by byaaasingle single singleODE ODE ODEisisisnew new newas as asisisisthe the thediscussion discussion discussionof of ofthe the thetotal total totalGaussian Gaussian Gaussiancurvature curvature curvatureof of ofaaasurface surface surfacedefined defined defined by by byaaapair pair pairof of ofODEs. ODEs. ODEs.

This This Thisvolume volume volumeisisisaaaprinted printed printedversion version versionof of ofaaawork work workthat that thatappears appears appearsin in inthe the theSynthesis Synthesis SynthesisDigital Digital DigitalLibrary Library Libraryof ofofEngineering Engineering Engineeringand and and Computer Computer ComputerScience. Science. Science.Synthesis Synthesis SynthesisLectures Lectures Lecturesprovide provide provideconcise, concise, concise,original original originalpresentations presentations presentationsof of ofimportant important importantresearch research researchand and anddevelopment development development topics, topics, topics,published published publishedquickly, quickly, quickly,in in indigital digital digitaland and andprint print printformats. formats. formats.For For Formore more moreinformation information informationvisit visit visitwww.morganclaypool.com www.morganclaypool.com www.morganclaypool.com

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Aspects of Differential Geometry II

Peter Peter Gilkey Gilkey JeongHyeong JeongHyeong Park Park Ramón RamónVázquez-Lorenzo Vázquez-Lorenzo

SSyntheSiS yntheSiS yntheSiS L LectureS ectureS ectureS on on on MatheMaticS atheMaticS atheMaticS and and and S StatiSticS tatiSticS tatiSticS M Steven Steven StevenG. G. G.Krantz, Krantz, Krantz,Series Series SeriesEditor Editor Editor

Aspects of Diﬀerential Geometry II

Synthesis Lectures on Mathematics and Statistics Editor Steven G. Krantz, Washington University, St. Louis

Aspects of Diﬀerential Geometry II Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo 2015

Aspects of Diﬀerential Geometry I Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo 2015

An Easy Path to Convex Analysis and Applications Boris S. Mordukhovich and Nguyen Mau Nam 2013

Applications of Aﬃne and Weyl Geometry Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2013

Essentials of Applied Mathematics for Engineers and Scientists, Second Edition Robert G. Watts 2012

Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations Goong Chen and Yu Huang 2011

Matrices in Engineering Problems Marvin J. Tobias 2011

e Integral: A Crux for Analysis Steven G. Krantz 2011

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Statistics is Easy! Second Edition Dennis Shasha and Manda Wilson 2010

Lectures on Financial Mathematics: Discrete Asset Pricing Greg Anderson and Alec N. Kercheval 2010

Jordan Canonical Form: eory and Practice Steven H. Weintraub 2009

e Geometry of Walker Manifolds Miguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2009

An Introduction to Multivariable Mathematics Leon Simon 2008

Jordan Canonical Form: Application to Diﬀerential Equations Steven H. Weintraub 2008

Statistics is Easy! Dennis Shasha and Manda Wilson 2008

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A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS Lecture #16 Series Editor: Steven G. Krantz, Washington University, St. Louis Series ISSN Print 1938-1743 Electronic 1938-1751

Aspects of Diﬀerential Geometry II Peter Gilkey University of Oregon, Eugene, OR

JeongHyeong Park Sungkyunkwan University, Suwon, Korea Institute for Advanced Study, Seoul, Korea

Ramón Vázquez-Lorenzo University of Santiago de Compostela, Santiago de Compostela, Spain

SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #16

M &C

Morgan & cLaypool publishers

ABSTRACT Diﬀerential Geometry is a wide ﬁeld. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition eorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. e exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. e de Rham cohomology of compact Lie groups and the Peter–Weyl eorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some diﬀerent proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface deﬁned by a pair of ODEs.

KEYWORDS Chern classes, Cliﬀord algebra, connection, de Rham cohomology, geodesic, Jacobi operator, Kähler geometry, Levi–Civita connection, Lie algebra, Lie group, Peter– Weyl eorem, pseudo-Riemannian geometry, Riemannian geometry, sheaf cohomology, simplicial cohomology, singular cohomology, symmetric space, volume of geodesic balls

vii

is book is dedicated to Alison, Arnie, Carmen, Junmin, Junpyo, Manuel, Montse, Rosalía, and Susana.

ix

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

4

Additional Topics in Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4.1 4.2 4.3 4.4

5

de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.1 5.2 5.3 5.4

6

Basic Properties of de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Cliﬀord Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 e Hodge Decomposition eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

7

Curves and Surfaces in Rn Given by ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Volume of Geodesic Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Holomorphic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Kähler Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 e Exponential Function of a Matrix Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 e Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Representations of a Compact Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bi-invariant pseudo-Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 e Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 e Classical Groups in Low Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 e Cohomology of Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 e Cohomology of the Unitary Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Homogeneous Spaces and Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1 7.2

Smooth Structures on Coset Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 e Isometry Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

x

7.3 7.4 7.5 7.6

8

e Lie Derivative and Killing Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 100 Local Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 e Global Geometry of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Other Cohomology eories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.1 8.2 8.3 8.4

Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Simplicial Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Singular Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

xi

Preface is two-volume series arose out of work by the three authors over a number of years both in teaching various courses and also in their research endeavors. e present volume (Book II) is comprised of ﬁve chapters that continue the discussion of Book I. In Chapter 4, we examine the geometry of curves which are the solution space of a constant coeﬃcient ordinary diﬀerential equation. We give necessary and suﬃcient conditions that the curves give a proper embedding and we examine when the total extrinsic curvature is ﬁnite. We examine similar questions for the total Gaussian curvature of a surface deﬁned by a pair of ODEs and apply the Gauss–Bonnet eorem to express the total Gaussian curvature in terms of the curves associated to the individual ODEs. We then examine the volume of a small geodesic ball in a Riemannian manifold. We show that if the scalar curvature is positive, then volume grows more slowly than it does in ﬂat space while if the scalar curvature is negative, then volume grows more rapidly than it does in ﬂat space. Chapter 4 concludes with a brief introduction to holomorphic and Kähler geometry. Chapter 5 treats de Rham cohomology. e basic properties are introduced and it is shown that de Rham cohomology satisﬁes the Eilenberg–Steenrod axioms; these are properties that all homology and cohomology theories have in common. We shall postpone until Chapter 8 a discussion of the Mayer–Vietoris sequence and the homotopy property as these depend upon some results in homological algebra that will be treated there. We determine the de Rham cohomology of the sphere and of real projective space. We introduce Cliﬀord algebras and present the Hodge Decomposition eorem. is is used to establish the Künneth formula and Poincaré duality. We treat the ﬁrst Chern class in some detail and use it to determine the ring structure of the de Rham cohomology of complex projective space. A brief introduction to the higher Chern classes and the Pontrjagin classes is given. Chapter 6 contains an introduction to the theory of Lie groups and Lie algebras. We restrict for the most part to matrix groups so that the exponential and log functions can be given explicitly in terms of convergent power series. We show in this setting that a closed subgroup of a matrix group is a Lie subgroup; this result is used to treat the geometry of the classic matrix groups (special linear group, orthogonal group in arbitrary signature, unitary group in arbitrary signature, symplectic group, etc.). If M is a compact Lie group with a bi-invariant metric, we show the one-parameter subgroups of the exponential map are geodesics. We express the de Rham cohomology of a compact connected Lie group in terms of the left-invariant diﬀerential forms and prove the Hopf structure theorem that shows the de Rham cohomology of a compact connected Lie group is a ﬁnitely generated exterior algebra on odd-dimensional generators. We use these results to determine the ring structure of the de Rham cohomology of the unitary group.

xii

PREFACE

We conclude Chapter 6 by discussing the orthogonality relations and the Peter–Weyl eorem which decomposes L2 as the direct sum of irreducible representations for a compact connected Lie group. Chapter 7 presents an introduction to the theory of homogeneous spaces and of symmetric spaces. We examine coset spaces and the group of isometries of a pseudo-Riemannian manifold. We introduce material related to the Lie derivative and Killing vector ﬁelds. We outline the geometry of homogeneous spaces, of local symmetric spaces, and of global symmetric spaces. Chapter 8 concludes Book II with a discussion of other cohomology theories. We introduce the necessary homological machinery to show that de Rham cohomology is a homotopy functor and has the Mayer–Vietoris long exact sequence; these two results also have a signiﬁcant geometric input. We relate simplicial cohomology, singular cohomology, and sheaf cohomology to de Rham cohomology. We have tried whenever possible to give the original references to major theorems in this area. We have provided a number of pictures to illustrate the discussion. Chapters 1 and 2 of Book I are suitable for an undergraduate course on “Calculus on Manifolds” and arose in that context out of a course at the University of Oregon. Chapter 3 is designed for an undergraduate course in Diﬀerential Geometry. erefore, Book I is suitable as an undergraduate text although, of course, it also forms the foundation of a graduate course in Diﬀerential Geometry as well. Book II can be used as a graduate text in Diﬀerential Geometry and arose in that context out of second year graduate courses in Diﬀerential Geometry at the University of Oregon and at Sungkyunkwan University. e material can, however, also form the basis of a second semester course at the undergraduate level as well. While much of the material is, of course, standard, many of the proofs are a bit diﬀerent from those given classically and we hope provide a new viewpoint on the subject. Our treatment of curves in Rm given by the solution to constant coeﬃcient ODEs which have ﬁnite total curvature is new as is the corresponding treatment of the total Gaussian curvature of a surface given by a pair of ODEs. ere are other examples; Diﬀerential Geometry is of necessity a vibrant and growing ﬁeld; it is not static! ere are, of course, many topics that we have not covered. is is a work on “Aspects of Diﬀerential Geometry” and of necessity must omit more topics than can be possibly included. For technical reasons, the material is divided into two books and each book is largely selfsuﬃcient. To facilitate cross references between the two books, we have numbered the chapters of Book I from 1 to 3, and the chapters of Book II from 4 to 8. Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo May 2015

xiii

Acknowledgments We have provided many images of famous mathematicians in these two books; mathematics is created by real people and we think having such images makes this point more explicit. e older pictures are in the public domain. We are grateful to the Archives of the Mathematisches Forschungsinstitut Oberwolfach for permitting us to use many images from their archives (R. Brauer, H. Cartan, S. Chern, G. de Rham, S. Eilenberg, H. Hopf, E. Kähler, H. Künneth, L. Nirenberg, H. Poincaré, W. Rinow, L. Vietoris, and H. Weyl); the use of these images was granted to us for the publication of these books only and their further reproduction is prohibited without their express permission. Some of the images (E. Beltrami, E. Cartan, G. Frobenius, and F. Klein) provided to us by the MFO are from the collection of the Mathematische Gesellschaft Hamburg; again, the use of any of these images was granted to us for the publication of these books only and their further reproduction is prohibited without their express permission. e research of the authors was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014053413) and by Project MTM2013-41335-P with FEDER funds (Spain). e authors are very grateful to Esteban Calviño-Louzao and Eduardo García-Río for constructive suggestions and assistance in proofreading. e assistance of Ekaterina Puﬃni of the Krill Institute of Technology has been invaluable. Wikipedia has been a useful guide to tracking down the original references and was a source of many of the older images that we have used that are in the public domain. Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo May 2015

1

CHAPTER

4

Additional Topics in Riemannian Geometry We continue the discussion of various topics in Riemannian geometry which was started in Chapter 3 of Book I. In Section 4.1, we discuss the geometry of curves and surfaces in Euclidean space that arise as solutions to an ordinary diﬀerential equation (ODE); this involves a nice application of the Gauss–Bonnet eorem that is typical in the subject. In Section 4.2, we relate the scalar curvature to the growth of the volume of geodesic balls. In Section 4.3, we turn to holomorphic geometry. We present the Cauchy–Riemann equations, deﬁne almost complex structures, and state the Newlander–Nirenberg eorem. We decompose d D @ C @N and introduce the spaces p;q . We deﬁne complex projective space. In Section 4.4, we present an introduction to Kähler geometry.

4.1

CURVES AND SURFACES IN Rn GIVEN BY ODES

Let P be a constant coeﬃcient ODE. e solution set of P deﬁnes a real analytic curve P .t/. Given two such ODEs, let ˙P1 ;P2 .t1 ; t2 / WD P1 .t1 / ˝ P2 .t2 / be a real analytic surface. In Section 4.1, we shall present some results of Gilkey et al. [21, 22] dealing with the total ﬁrst curvature ŒP of the curve P and the total Gaussian curvature KŒ˙P1 ;P2 of the surface ˙P1 ;P2 . e study of KŒ˙P1 ;P2 illustrates the manner in which the Gauss–Bonnet eorem is often used. Let P ./ WD .n/ C cn 1 .n 1/ C C c0 for ci 2 R be a constant coeﬃcient ODE. Let P D PP be the associated characteristic polynomial and let R D RP be the roots of P :

P ./ WD n C cn 1 n

1

C C c0

and R WD f 2 C W P ./ D 0g :

We suppose that all the roots of P have multiplicity 1; the extension to the higher multiplicity case is for the most part straightforward modulo adding suitable powers of t . Enumerate the roots R of P in the form:

R D fr1 ; : : : ; rk ; z1 ; zN1 ; : : : ; zu ; zNu g for k C 2u D n ; p where ri 2 R for 1 i k and where zj D aj C 1bj with bj > 0 for 1 j u. Choose the labeling so r1 > r2 > and a1 a2 . e standard basis for the solution space of P

2

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

is given by the real analytic functions 1 WD e r1 t ; : : : ; k WD e rk t ; kC1 WD e a1 t cos.b1 t/; kC2 WD e a1 t sin.b1 t /; :::; au t au t n 1 WD e cos.bu t/; n WD e sin.bu t/ :

e associated curve P W R ! Rn , the curvature of P , the element of arc length ds , and the total ﬁrst curvature ŒP are given by: kP P ^ R P k P .t/ WD .1 .t/; : : : ; n .t //; WD ; kP P k3 Z Z 1 kP P ^ R P k ds WD kP P kdt; ŒP WD ds D dt : kP P k2 1 P

Let 0 > rk and if r1 > rk for any 2 R fr1 ; rk g. is means that lim t !1 ke r1 t P k D 1, lim t ! 1 ke rk t P k D 1, and lim t! 1 kP k D 1. e remaining roots are then said to be subdominant. We refer to P. Gilkey, C. Y. Kim, H. Matsuda, J. H. Park, and S. Yorozu [21] for the proof of the following result.

C. Y. Kim

H. Matsuda

S. Yorozu

eorem 4.1 Let P be the curve deﬁned by a real constant coeﬃcient ordinary diﬀerential equation P of order n with simple roots and with r1 > 0 > rk .

1. e curve P is a proper embedding of R into Rn with inﬁnite length. e total ﬁrst curvature ŒP is ﬁnite if and only if the real roots are dominant. 2. If there are no complex roots, then ŒP 4 n.n 1/. Given > 0, there exists an ODE P of order n with no complex roots so that ŒP 13 .n 1/ and so that the real roots are dominant. Consequently, any universal upper bound must grow at least linearly with n and at worst quadratically. If subdominant complex roots are allowed, then there is no uniform upper bound. If P is deﬁned by some other basis for the solution space of P than the standard basis, then Assertion 1 continues to hold and can be generalized to the case when the roots have multiplicities greater than 1.

4.1. CURVES AND SURFACES IN Rn GIVEN BY ODES

Given a constant coeﬃcient ODE P1 (resp. P2 ) of order n1 (resp. n2 ), we can deﬁne ˙.t1 ; t2 / WD P1 .t1 / ˝ P2 .t2 / mapping R2 to Rn1 n2 . If fi g is the standard basis for the solution space of P1 and if f j g is the standard basis for the solution space of P2 , then fi .t1 / j .t2 /g are the coordinates of ˙.t1 ; t2 /. eorem 4.1 extends to this setting where we R replace by the Gaussian curvature K . e total absolute Gaussian curvature is deﬁned to be ˙ jKj j dvol j. We refer to Gilkey et al. [22] for the proof of the following result. eorem 4.2

Adopt the notation established above.

1. If the real roots of P1 and P2 are dominant, then ˙ is a geodesically complete proper embedding of R2 into Rn1 n2 of ﬁnite total absolute Gaussian curvature which has inﬁnite volume. 2. Let H be the mean curvature vector. Assume all the roots of P1 (resp. P2 ) are real and that there are at least two positive and at least two negative roots. en there exist D .˙ / > 0 and C D C.˙/ > 0 so that kH k C e k.t1 ;t2 /k . Furthermore, kH k 2 L3 .j dvol j/. Finally, given > 0, there exists ˙ of this form so kH k … L3 .j dvol j/. Consequently, p D 3 is the optimal universal index. If there are complex roots of P1 or of P2 which are dominant, then jKjŒ˙ can be inﬁnite. erefore, the assumption that the real roots are dominant is necessary in Assertion 1. It is possible to reduce the computation of KŒ˙ to a question on the associated curves P1 and P2 using the Gauss–Bonnet eorem. Let be the curve deﬁned by an ODE with simple roots and dominant real roots. Set ..t P / ^ .t /; P .t / ^ R .t // : kP .t / ^ .t /k kP .t /k3

.t/ WD

We apply the Cauchy–Schwarz inequality to see j .t/j .t / and hence ds is integrable and we may set: Z 1 Z 1 .P .t / ^ .t /; P .t / ^ .t R // Œ WD ds D dt : P .t / ^ .t /k kP .t /k2 1 1 k Let ˙r .t/ WD ˙.t; ˙r/ D 1 .t / ˝ 2 .˙r/. Let g be the geodesic curvature deﬁned by the inward pointing unit normal. If all the roots of P1 and of P2 are simple and if the real roots are dominant, then: Z r g . ˙r /.t/ds D Œ1 . lim Lemma 4.3

r!1

r

Proof. Let ˙= i WD @ ti ˙ and ˙= ij WD @ ti @ tj ˙ . We will use the inward unit normal to apply the Gauss–Bonnet eorem. is points in the direction of f˙=1 ^ ˙=2 .t; ˙r/g. One has: g .t; ˙r/ds D f.˙=1 ^ ˙=2 ; ˙=1 ^ ˙=11 / g

1

k˙=1 k

2

g.t; ˙r/dt :

3

4

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

P P First let t2 D r . Decompose 1 .t1 / D i e ri t1 ei and 2 .t2 / D j e sj t2 fj relative to bases fei g for Rn1 and ffj g for Rn2 . We express 2 .t2 / D e s1 t2 .f1 C E .t2 // where the remainder E .t2 / is exponentially suppressed, i.e., satisﬁes an estimate of the form kE .t2 /k e t2 for some > 0 if t2 >> 0. In this setting, we shall write 2 .t2 / e s1 t2 f1 . We compute: ˙=1 P 1 ˝ e s1 r f1 ; ˙=2 1 ˝ s1 e s1 r f1 ; 2s1 r g D k˙=1 ^ ˙=2 k js1 je kP 1 ^ 1 k; ˙=11 R 1 ˝ e s1 r f1 ;

g . r /ds D

.˙=1 ^ ˙=2 ; ˙=1 ^ ˙=11 /g 1 k˙=1 k 2 dt s1 e 4s1 r .P 1 .t1 / ^ 1 .t1 /; P 1 .t1 / ^ R 1 .t1 // dt : js1 je 4s1 r k1 .t1 / ^ P 1 .t1 /k kP 1 .t1 /k2

is gives .1 /dt in the limit since s1 > 0. We do not need to change the sign of the normal but again get a negative sign if sk < 0 since C jsskk j D 1. t u eorem 4.4

If the roots of Pi are simple and if the real roots are dominant for i D 1; 2, then:

1. 0 D KŒ˙

2Œ1

2Œ2 C 2 .

2. If there are no complex roots, then jKŒ˙ j

fn1 .n1 2

1/ C n2 .n2

1/g C 2 .

3. If P1 and P2 are second order ODEs, then KŒ˙ D 0. Proof. We apply the Gauss–Bonnet eorem to the square ˙.Œ r; r Œ r; r/. Let ˛i be the interior angles. We then have: 2

D

Z

r

Z

r

r

C

Z

r

4 X K.t1 ; t2 /gdt1 dt2 C .

r r

g .˙.t; r//ds C

Z

iD1 r r

˛i / C

Z

r

g .˙.t; r//ds r

g .˙.r; t //ds C

Z

r

g .˙. r; t //ds : r

Let ˛1 be the angle at ˙.r; r/. Since ˙=1 .r; r/ r1 ˙.r; r/ and since ˙=2 .r; r/ s1 ˙.r; r/, ˙=1 and ˙=2 point in approximately the same direction. Consequently, cos.˛1 / 1 and ˛1 0. Keeping careful track of the signs shows the other angles also are close to 0. Assertion 1 then follows from Lemma 4.3. Since jŒi j Œi , Assertion 2 follows from Assertion 1 and from Assertion 2 of eorem 4.1. To prove Assertion 3, we make a direct computation. Let .t/ WD .e at ; e bt /. en Z 1 Z 1 j.a b/abje .aCb/t j.a b/abje .a b/t Œ D dt D dt : 2 2at C b 2 e 2bt 2 2.a b/t C b 2 1 a e 1 a e We have a

b > 0. We change variables setting x WD e .a b/t to express Z 1 Z 1 jaj jabj 1 dx D dx : Œ D 2x2 C b2 a2 2 a jbj 0 0 2x C 1 b

4.2. VOLUME OF GEODESIC BALLS

We again change variables setting y D

jaj x jbj

Œ D

Z

to express 1

0

y2

1 dy D : C1 2

Assertion 3 now follows from Assertion 1. 4.1.1 EXAMPLE. Let 1 .t1 / D .e 2t1 ; e gK D

9e 10t1

3t1

t u / and 2 .t2 / D .e 2t2 ; e

3t2

/. We may compute:

125e 4.t1 Ct2 / . 9 C 4e 10.t1 Ct2 / / : C 9e 10t2 C e 10.t1 Ct2 / C 4e 10.2t1 Ct2 / C 4e 10.t1 C2t2 /

Let D be the denominator. en

D e 10.2t1 Ct2 / D 2 e 10.3t1 C3t2 /

and D e 10.t1 C2t2 / so and D e 15t1 C15t2 :

e numerator is bounded by e 14.t1 Ct2 / . us, gK is integrable. Let .t / WD e 10.t1 Ct2 / . e Gaussian curvature changes sign. It is positive for 94 and negative for .t1 ; t2 / < 49 . It does not vanish identically and Assertion 2 of eorem 4.4 is non-trivial. If we set 1 .t1 / D .e 2t1 ; 1; e 3t1 / and 2 .t2 / D .e 2t2 ; e 3t2 /, then Z Z gK dvol :951333 and jgKj dvol 1:09409 : ˙

˙

us, the total Gaussian curvature is non-zero for this example and again K changes sign.

4.2

VOLUME OF GEODESIC BALLS

We recall some notation established previously in Book I. Let g be a pseudo-Riemannian metric on a smooth manifold M of dimension m; g is a smooth section to S 2 .T M / and can be regarded as a smooth family of non-degenerate symmetric bilinear forms on each tangent space TP M . Let r be the Levi–Civita connection; it is characterized by the properties: rX Y

rY X D ŒX; Y

and X.g.Y; Z// D g.rX Y; Z/ C g.Y; rX Z/ :

is connection was ﬁrst studied by the Italian mathematician Tullio Levi–Civita.

T. Levi–Civita (1873–1941)

5

6

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

Let R and j dvol j be the curvature operator and the Riemannian measure, respectively:

R.X; Y / WD rX rY

rY rX

rŒX;Y

1

and j dvol j D j det.gij /j 2 dx 1 dx m :

4.2.1 THE CURVATURE TENSOR IN GEODESIC COORDINATES. Let expP be the exponential map deﬁned by g . It is a diﬀeomorphism from a neighborhood of 0 in TP M to a neighborhood of P in M which is characterized by the property that the curve v W t ! expP .t v/ is a geodesic with v .0/ D P and P v .0/ D v for any v 2 TP M . Let logP be the local inverse. If eE WD fe1 ; : : : ; em g is a basis for TP M , let .x/ E WD expP .x 1 e1 C C x m em / :

is gives a system of local coordinates on M called geodesic coordinates. We adopt the Einstein convention and sum over repeated indices where one index is up and one index is down and set, for example, x i ei WD x 1 e1 C C x m em . Let i; j; k; ` be indices with 1 i; j; k; ` m. Let: gij WD [email protected] i ; @x j /; gij=k WD @x k gij ; gij=k` WD @x ` @x k gij ; Rij k` WD [email protected] i ; @x j /@x k ; @x ` / :

In a system of geodesic coordinates, we can express the components of the curvature tensor in terms of the 2-jets of the metric, and we can express the 2-jets of the metric in terms of the components of the curvature tensor at the center of the coordinate system. Let P be a point of a pseudo-Riemannian manifold .M; g/ and let xE be a system of geodesic coordinates centered at P . We have:

Lemma 4.5

gij=k .P / D 0; gi k=j ` .P / D .gkj= i ` C gk`= ij /.P / D gj `= i k .P /; Rij k` .P / D gi k=j ` .P / gi`=j k .P /; 3gi k=j ` .P / D Rij k` .P / Ri`j k .P / :

Proof. If s , u, v , and w are vectors in Rm , denote the corresponding (constant) coordinate vector ﬁelds by: S./ D s i @x i ;

U./ D ui @x i ;

V ./ D v i @x i ;

and W ./ D w i @x i :

e bracket of any pair of these vector ﬁelds vanishes. We have g.rV V; W / D Vg.V; W /

1 W g.V; V 2

/:

4.2. VOLUME OF GEODESIC BALLS

e curve t ! tv is a geodesic so g.rV V; W /.t v/ D 0 for t 2 Œ0; 1/. Consequently, 0 D 2Vg.V; W /.P / W g.V; V /.P / and 0 D 2V Vg.V; W /.P / V W g.V; V /.P / :

(4.2.a) (4.2.b)

We set V D W in Equation (4.2.a) to see Vg.V; V /.P / D 0. Let V ."/ WD V C "W . e identity @ fV ./g.V ./; V .//.P /gjD0 D 0 implies that (4.2.c)

0 D W g.V; V /.P / C 2Vg.V; W /.P / :

We use Equations (4.2.a) and (4.2.c) to see that (4.2.d)

0 D W g.V; V /.P / :

We polarize this identity. Let V ."/ WD V C "U . We diﬀerentiate the identity of Equation (4.2.d), and set " D 0 to see 0 D W g.U; V /.P / so gij=k .P / D 0 :

We set V D W in Equation (4.2.b) to see that 0 D V Vg.V; V /.P /. We polarize this identity. Let V ."/ WD V C "W . We diﬀerentiate the identity 0 D V ./V ./g.V ./; V .//.P / with respect to " and set " D 0 to see 0 D V Vg.V; W /.P / C V W g.V; V /.P /. We use this identity and Equation (4.2.b) to see: 0 D V Vg.V; W /.P /

and 0 D V W g.V; V /.P / :

(4.2.e)

We polarize these identities. Let V ."/ WD V C "U . We diﬀerentiate the identities of Equation (4.2.e) with respect to " and set " D 0 to see that 0 D 2U Vg.V; W /.P / C V Vg.U; W /.P /; 0 D U W g.V; V /.P / C 2V W g.U; V /.P / :

(4.2.f )

Note that V Vg.U; W / is symmetric in U and W . We use the relations of Equation (4.2.f) twice to see that for all U , V , W : V Vg.U; W /.P / D

2U Vg.V; W /.P / D

2W Vg.U; V /.P / D U W g.V; V /.P / :

Let V ."/ WD V C "S . We diﬀerentiate this identity with respect to " and set " to 0 to see V Sg.U; W /.P / D USg.W; V /.P / U Vg.W; S /.P / D U W g.V; S /.P / gik=j ` .P / D .gkj= i ` C gk`= ij /.P / D gj `= i k .P / :

so

Because gab=c .P / D 0, we have that Rij k` .P / D 12 fgj `= i k C gi k=j `

gj k= i `

gi `=j k g.P / D gik=j ` .P /

gi `=j k .P / :

We complete the proof by using the identities we have already derived to see that Rij k` .P / Ri`j k .P / D fgik=j ` gi `=j k gij=k` C gik=j ` g.P / D 3gi k=j ` .P / :

7

8

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

4.2.2 THE GROWTH OF VOLUME OF BALLS. e scalar curvature is deﬁned by setting WD g j k Rij k i ; it plays a crucial role in the Gauss–Bonnet eorem. It is also closely related to the volume of small geodesic balls. Let P be a point of a Riemannian manifold of dimension m. m Let VrM .P / be the volume of the geodesic ball BrM .P / of radius r about P . Let VrR be the volume of the metric ball of radius r about the origin in Rm . m

As r # 0, we have VrM D VrR

eorem 4.6

n

1

o C O.r 4 / . r 2 6.mC2/

We remark that there is a full normalized Taylor series in r 2 and refer to Gray [23] for further details. If D g ij Iij , then the next term in the expansion is: 3kRk2 C 8kk2 C 5 2 C 18 : 360.m C 2/.m C 4/

r4

Proof. We work in geodesic polar coordinates. We expand gij .x/ E D ıij C 12 gij=k` x k x ` C O.kxk E 4 /; X 1 det.gij /.x/ E D1C gi i=k` x k x ` C O.kxk E 4 /; 2 i;k;`

1X det.gij /.x/ E D1C gi i=k` x k x ` C O.kxk E 4 /; 4 i;k;` 8 9 Z < = X 1 VrM D gi i=k` x k x ` C d xE : 1C ; 4 kxk E 2 r : 1 2

i;k;`

e integral of the cubic term vanishes and Z Z

d xE

D

x 1 x 1 d xE

D

kxk E 2 r kxk E 2 r

D VrM

D D

Z

r

Z

R

x k x ` D 0 for k ¤ `. We compute:

%m

1

j dvol j. /d% D Vol.S m 1 /r m =m; %D0 Z Z Z 1 1 r 2 kxk E d xE D %m 1 %2 j dvol j./d% m kxk m %D0 2S m 1 E 2 r 1 Vol.S m 1 /r mC2 ; m.m C 2/ 8 9 < = X 1 m 1 r Vol.S m 1 / 1 C r 2 gi i=jj C : ; m 4.m C 2/ i;j 1 m VrR 1 r 2 Rijj i C : 6.m C 2/ 2S m 1

t u

4.3. HOLOMORPHIC GEOMETRY

4.3

HOLOMORPHIC GEOMETRY

We say that a function f .z/ D f .x; y/ from an open subset O of C D R2 to C D R2 is holomorphic if it is complex diﬀerentiable, i.e., if f .z C ı/ ı ı!0

f 0 .z/ D lim

f .z/

exists and is independent of the particularpcomplex direction ı which is used to approach the origin for every z 2 O. Decompose f D u C 1v into real and imaginary parts. Letting ı approach 0 along the real or the complex axis then yields the Cauchy–Riemann Equations [13, 54] @x u D @y v

and @x v D @y u :

(4.3.a)

Holomorphic functions have many properties; the sum, product, diﬀerence, quotient (if appropriate), and composition of holomorphic functions is holomorphic. e Taylor series of a holomorphic function converges uniformly to the holomorphic function near each point in the domain; conversely, since the uniform limit of holomorphic functions is again holomorphic, a complexvalued function of a complex variable which is given by a convergent Taylor series is necessarily holomorphic. A vector-valued function F W O C n ! C p is said to be holomorphic on O if each of the components of F is holomorphic in each of the variables separately. e complex Jacobian is the matrix Fc0 WD @z ˛ Fˇ . If n D p and if det.Fc0 / ¤ 0, then F is a local diﬀeomorphism and a local inverse F 1 is again holomorphic. If f W O C n ! C is a scalar non-constant holomorphic function and if O is connected, then fz W f .z/ ¤ 0g is again connected. An almost complex structure on a smooth manifold M is a smooth endomorphism J of TM so that J 2 D Id. A coordinate system .x ˛ ; y ˛ / is said to be a holomorphic coordinate system if J @x ˛ D @y ˛ and J @y ˛ D @x ˛ . e Nijenhuis tensor NJ [47] is given by: NJ .X; Y / WD ŒX; Y C J ŒJX; Y C J ŒX; J Y

ŒJX; J Y :

e complex tangent bundle is the eigen-subbundle of the complexiﬁed tangent bundle deﬁned by setting: p TC .M / D fv 2 TM ˝R C W J v D 1vg : We extend the Lie bracket from real tangent bundle TM to the complexiﬁed tangent bundle TM ˝R C to be complex bilinear. We say that TC .M / is closed under bracket if given sections and in C 1 TC .M /, we have that the bracket Œ; belongs to C 1 TC .M / as well. We refer to Newlander and Nirenberg [46] for the proof of the following result which is called the Newlander– Nirenberg eorem.

9

10

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

Louis Nirenberg (1925–) e following is a deep result in the theory of partial diﬀerential equations which is beyond the scope of this book and which can be regarded as a complex version of the real Frobenius eorem (see eorem 2.8 in Book I).

Let .M; J / be an almost complex manifold of dimension m D 2m N . e following conditions are equivalent and if any is satisﬁed, then .M; J / is said to be a complex manifold and the structure J is said to be an integrable complex structure: eorem 4.7

1. ere are holomorphic coordinate charts covering M . 2. e Nijenhuis tensor is 0. 3. e complex tangent bundle TC .M / is closed under bracket. 4.3.1 ISOTHERMAL COORDINATES. Recall that a system of local coordinates .x; y/ on a Riemann surface is said to be isothermal if ds 2 D e 2h .dx 2 C dy 2 / for some smooth function h. Corollary 4.8

ere exist isothermal coordinate charts covering any Riemann surface.

Proof. Let P be a point of a Riemann surface M . Let fe 1 ; e 2 g be a local orthonormal frame for TM which is deﬁned near P . en the canonical almost complex structure Je 1 D e 2

and Je 2 D e 1

is integrable since TC M is 1-dimensional. erefore, we can choose local holomorphic coordinates so J @x D @y and J @y D @x . Since J is unitary, J g D g . Consequently, [email protected] ; @x / D [email protected] ; @y /

and [email protected] ; @y / D 0

so g D e 2h .dx 2 C dy 2 / for some smooth conformal factor h.

Let .M; J / be a complex manifold. e transition functions relating any two holomorphic coordinate systems are holomorphic. Conversely, if .M; J / is a manifold which admits a cover by charts such that the transition functions are holomorphic, then we can recover J by setting J @x ˛ D @y ˛

and J @y ˛ D @x ˛ :

4.3. HOLOMORPHIC GEOMETRY

11

Equation (4.3.a) then ensures that J is well-deﬁned and independent of the particular local coordinate system chosen. Let O˛ and Oˇ be open subsets of C mN and let F W O˛ ! Oˇ be a holomorphic diﬀeomorphism. Let Fc0 WD @z ˛ Fˇ be the holomorphic Jacobian and let F 0 be the ordinary Jacobian. e Cauchy–Riemann equations imply det.F 0 / D j det.Fc0 /j2 > 0. Consequently, every holomorphic manifold is orientable. 4.3.2 THE OPERATORS @ AND @N . Let .M; J / be a complex manifold and let m N .x 1 ; : : : ; x mN ; y 1 ; : : : ; yp / be a system of local holomorphic coordinates. Introduce holomorphic variables z ˛ WD x ˛ C 1y ˛ and deﬁne p p dz ˛ D dx ˛ C 1dy ˛ ; d zN ˛ D dx ˛ 1dy ˛ ; p p [email protected] ˛ /; @zN ˛ WD 12 [email protected] ˛ C [email protected] ˛ / : @z ˛ WD 12 [email protected] ˛

Extend the usual pairing h; i between a cotangent and a tangent vector to be complex bilinear. We then have: hdz ˛ ; @z ˇ i D hd zN ˛ ; @zN ˇ i D ıˇ˛

and hdz ˛ ; @zN ˇ i D hd zN ˛ ; @z ˇ i D 0 :

Let f be a smooth complex-valued function. We sum over repeated indices to deﬁne: @f WD [email protected] ˛ f /dz ˛

N WD [email protected] ˛ f /d zN ˛ : and @f

e Cauchy–Riemann equations show that a smooth complex-valued function f is holomorN D 0. If I D f1 i1 < < ip mg phic if and only if @f N and J D f1 j1 < jq mg N are collections of indices, set: dz I WD dz i1 ^ ^ dz ip

and

d zN J WD d zN j1 ^ ^ d zN jq :

N , and Let p;q WD spanjI jDp;jJ jDq dz I ^ d zN J . e Cauchy–Riemann equations show that @f , @f p;q are invariantly deﬁned, i.e., independent of the particular holomorphic coordinate system which is chosen. We may decompose the complex exterior algebra in the form: n .M / ˝R C D ˚pCqDn p;q M :

We extend @ and @N to maps @.fI;J dz I ^ d zN J / D @fI;J ^ dz I ^ d zN J W C 1 .p;q M / ! C 1 .pC1;q M / and N I;J dz I ^ d zN J / D @f N I;J ^ dz I ^ d zN J W C 1 .p;q M / ! C 1 .p;qC1 M / : @.f

Again, these are independent of the particular holomorphic coordinate system chosen. We have: N N D 0. d D @ C @; @@ D 0; @N @N D 0; @@N C @@

12

4. ADDITIONAL TOPICS IN RIEMANNIAN GEOMETRY

4.3.3 COMPLEX PROJECTIVE SPACE. We continue the discussion of Section 2.3.6 of Book I. Let CPm WD f 2 MmC1 .C/ W D ; 2 D ; Rank./ D 1g

be the set of orthogonal projections of rank 1 in MmC1 .C/; CPm is a compact metric space. Let CP m be the set of complex lines in C mC1 through the origin, i.e., 1-dimensional subspaces of C mC1 . If 2 CP m , let be orthogonal projection on ; this identiﬁes CP m with CPm and gives CP m the structure of a compact metric space. e non-zero complex numbers C f0g act on C mC1 f0g by scalar multiplication; this action restricts to an action of S 1 on S 2mC1 . e map W z ! z˚ C deﬁnes a homeomorphism from S 2mC1 =S 1 with the quotient topology to ˚ m mC1 CP or from C f0g = C f0g with the quotient topology to CP m . We give CP m the structure of a holomorphic manifold as follows. Let Ui WD fz 2 C mC1 W z i ¤ 0g

be the associated open covers of C mC1

and fOi WD .Ui /g

f0g and CP m , respectively. Let

Fi .w 1 ; : : : ; w m / D .w 1 ; : : : ; 1; : : : ; w m / W C m ! Ui

be deﬁned by putting a 1 in the i th position. en ı Fi is a homeomorphism from C m onto Oi that deﬁnes a local coordinate chart. More speciﬁcally, if z 2 Oi , let zij .z/ WD z j =z i . Since zij .z/ D zij .z/ for ¤ 0, the zij descend to deﬁne continuous functions on Oi which give local coordinates (called homogeneous coordinates) with ( ) wj for j < i j zi .Fi w/ D : w j 1 for j > i Since zij D zkj =zki , the transition functions are holomorphic so CP m has the structure of a complex manifold and ı Fi is a holomorphic diﬀeomorphism from C m to Oi . A point of CP m is a line in C mC1 . e ﬁber of the tautological line bundle L is deﬁned to be that line, i.e., L WD f z 2 CP m C mC1 W z 2 g :

(4.3.b)

We deﬁne local sections to L over the open charts Oi by setting: si ./ WD .; zi1 ./; : : : ; zimC1 .//

for

2 Oi :

Since si ./ D zij ./sj ./ on Oi \ Oj , L is a holomorphic line bundle over CP m . If m D 1, then we have two coordinate systems .C; z1 / and .C; z2 / and the transition rule is z D w 1 on C f0g. is is the Riemann sphere so CP 1 D S 2 . is can also be seen combinatorially. Let .w1 ; w2 / be the usual coordinates on C 2 . Deﬁne F .w1 ; w2 / WD .2 0. m 3. If ! is a non-vanishing m-form on M , then 0 ¤ Œ! 2 HdR .M /.

R Proof. Let R R ! be a smooth m-form with M ! ¤ 0. If ! D d !Q , then Stokes’ eorem implies m Q D 0 which is false as, by assumption, @M D ;. Consequently, Œ! ¤ 0 in HdR .M / M ! D @M ! which proves Assertion 1. Put a Riemannian metric R on M and let dvol be the oriented volume form. is is a non-vanishing m-form on M with M dvol D Vol.M /. Assertion 2 now follows

5.1. BASIC PROPERTIES OF DE RHAM COHOMOLOGY

21

from Assertion 1. If ! is an m-form, we can express ! D f dvol. By eorem 5.1, we may suppose that M is connected; either f is always positive or R hence Rsince ! is non-vanishing, R m f is always negative. Since M ! D M f j dvol j, M ! ¤ 0 and hence 0 ¤ Œ! 2 HdR .M / by Assertion 1. t u We can now compute the de Rham cohomology groups of the unit sphere S m in RmC1 . Let xE D .x 1 ; : : : ; x mC1 / be the usual coordinates on RmC1 . Deﬁne !m on the punctured Euclidean space RmC1 f0g by setting: !m D kxk E

m 1

mC1 X iD1

b

. 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1 :

b

e notation d x i indicates that this element is to be deleted from the wedge product. 8 9 if p D 0 > ˆ = < R Œ1 p p eorem 5.4 If m 1, HdR .RmC1 f0g/ D HdR .S m / D R Œ!m if p D m . ˆ > : ; 0 otherwise Proof. Let N be the north pole and let S be the south pole of S m . Let

O1 WD S m

O2 D S m

and

fN g

fSg :

Since Oi is the punctured sphere, Oi is homotopy equivalent to a point and O1 \ O2 is diﬀeomorphic to S m 1 .0; / which is homotopy equivalent to S m 1 . erefore, the Mayer–Vietoris sequence yields: p 1 p 1 p 1 p p p HdR .O1 / ˚ HdR .O2 / ! HdR .O1 \ O2 / ! HdR .S m / ! HdR .O1 / ˚ HdR .O2 /

jo

p 1 p 1 HdR .pt/ ˚ HdR .pt/

jo

p 1 HdR .S m

1

jo

p p HdR .pt/ ˚ HdR .pt/ :

/

p p p 1 Since HdR .pt/ D 0 for p > 0, we conclude HdR .S m / ' HdR .S m 1 / for p 2. We now examine the start of the sequence: 0 0 ! R ! R ˚ R ! HdR .S m

1

v

1 / !HdR .S m / ! 0 :

e sequence simpliﬁes to become: 0 0 ! R ! HdR .S m

1

v

1 / !HdR .S m / ! 0 :

0 If m D 1, then S 0 is the disjoint union of two points so HdR .S 0 / D R ˚ R. Consequently, 1 0 1 HdR .S 1 / D R. If m 2, then HdR .S m 1 / D R and we conclude HdR .S m / D 0. is establishes m m the additive structure of HdR .S /. Since the sphere S is a deformation retract of RmC1 f0g, p p we may use eorem 5.2 to see HdR .S m / D HdR .RmC1 f0g/. erefore, we may identify these m two groups. We must now show that the generator of HdR .S m / is Œ!m . We show that d!m D 0 by computing:

22

5. DE RHAM COHOMOLOGY m 1

d!m D d fkxk E Ckxk E D

d

iD1

b

. 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1

mC1 X iD1

.m C 1/kxk E Ckxk E

D

m 1

mC1 X

g^

m 3

mC1 X kD1

k

k

x dx ^

mC1 X iD1

iD1

.m C 1/kxk E C.m C 1/kxk E

b

. 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1

mC1 X

m 1

b

. 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1

b

. 1/iC1 dx i ^ dx 1 ^ ^ dx i ^ ^ dx mC1

m 3 m 1

kxk E 2 dx 1 ^ ^ dx mC1

dx 1 ^ ^ dx mC1 D 0 .

Since kxk E D 1 on S m , we use Stokes’ eorem to compute: Z

Sm

!m

D D D

Z

Sm

mC1 X iD1

b

. 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1

D mC1

(mC1 ) X . 1/iC1 x i dx 1 ^ ^ dx i ^ ^ dx mC1 d

D mC1

.m C 1/dx 1 ^ ^ dx mC1 D .m C 1/ vol.D mC1 / ¤ 0 :

Z Z

iD1

By eorem 5.3, Œ!m ¤ 0 in H m .S m / D H m .RmC1

b

f0g/.

t u

e outward unit normal on S m is given by setting WD x i @x i . Let int be interior multiplication – see Section 5.2 for details. We then have !m jS m D int./.dx 1 ^ ^ dx mC1 / :

e construction is invariant under the action of the special orthogonal group SO.m C 1/. Consequently, !m is an SO.m C 1/ invariant m-form on S m . Consequently, !m is a constant multiple of the oriented volume form on S m . One evaluates at the north pole to see that the multiple is 1 so in fact !m is the oriented volume form. If r W RmC1 f0g ! S m is the radial retraction xE ! kxk E 1 xE , then on RmC1 f0g we have !m D r .!m jS m /. Let a.x/ E WD xE be the antipodal map of the sphere. Let Z2 WD fId; ag act smoothly without ﬁxed points on the sphere S k for k 1. e quotient S m =Z2 is the real projective space RP m . We study the behavior of de Rham cohomology under ﬁnite regular coverings and compute the cohomology of real projective space as an application.

5.1. BASIC PROPERTIES OF DE RHAM COHOMOLOGY

23

eorem 5.5

1. If is a ﬁnite group which acts smoothly and without ﬁxed points on a smooth manifold M , we can let MQ WD M= and let W M ! MQ be the associated regular covering projection. en W H p .MQ / ! H p .M / is injective and dR

dR

p rangef g D fŒ 2 HdR .M / W Œ D Œ for all 2 8 9 R if p D 0 ˆ > < = p m 2. HdR .RP / D R if p D m and m is odd . ˆ > : ; 0 otherwise

g.

3. RP m is orientable if and only if m is odd. Proof. Since W M ! MQ is a covering projection, it is a local diﬀeomorphism. Consequently, if Q 2 C 1 .p MQ /, then Q is invariant under the action of the deck group . e reverse implication also follows and, consequently: fC 1 .p MQ /g D f 2 C 1 .p M / W D for all 2

g:

Q D 0 in H p .M /. at means there exists Let Q 2 C 1 .p MQ / satisfy d Q D 0. Suppose Œ dR so that Q D d . Since ı D for any 2 , D . is shows that Q D

If ı 2

1 X Q 1 X

D

d D d1 j j j j

2

2

for

1 WD

1 X

: j j

2

, then ı 1 D

1 X 1 X 1 X ı D . ı/ D

D 1 j j j j j j

2

2

2

since the elements ı as ranges through also parametrize the elements of . Consequently, 1 is invariant under the action of the deck group and can be written in the form 1 D Q 1 . is implies .Q d Q 1 / D 0. Since is a local diﬀeomorphism, Q d Q 1 D 0 so Œ D 0 in p HdR .MQ /. is shows is injective on cohomology; a similar argument averaging over the deck group shows rangef g is the invariant cohomology. is proves Assertion 1. We specialize to the natural projection W S m ! RP m . Let !m be as deﬁned in eorem 5.4 and let a be the antipodal map. We may compute that a !m D . 1/mC1 !m . By Asserp p tion 1, HdR .S m / D 0 for p ¤ 0; m. erefore, HdR .RP m / D 0 in that range. Furthermore, since m 0 m m S is arc connected, RP is arc connected so HdR .RP m / D R. Finally, HdR .RP m / is isomorphic to the invariant cohomology elements of H m .S m /. Since a !m D . 1/mC1 !m and Œ!m m m generates HdR .S m /, HdR .RP m / vanishes if m is even and is R if m is odd. If m is odd, !m is

24

5. DE RHAM COHOMOLOGY

invariant under the antipodal map and descends to deﬁne an orientation of Rm . If m is even, we m apply eorem 5.3 to see RP m is not orientable since HdR .RP m / D f0g. Assertion 3 also follows from Lemma 2.18 in Book I. We now determine the additive structure of the de Rham cohomology of complex projective space. We shall determine the ring structure in eorem 5.18 but postpone that discussion until after we have introduced the ﬁrst Chern class. ( ) R if 0 p 2k and p is even p k eorem 5.6 If k 1, then HdR .CP / D . 0 otherwise Proof. Since CP 1 D S 2 , eorem 5.4 establishes the desired result if k D 1. erefore, we proceed by induction to establish the result in general. We have CP k WD S 2kC1 =S 1 . Let zE D .z 1 ; w/ E 2 S 2kC1

for

z1 2 C

and w E 2 Ck :

We deﬁne: E : U1 WD f.z 1 ; w/ E 2 S 2kC1 W jz 1 j < 2jwjg E and U2 WD f.z 1 ; w/ E 2 S 2kC1 W jz 1 j > 21 jwjg

ese are open subsets of S 2kC1 which are invariant under the action of S 1 ; the two open sets Oi WD Ui =S 1 form an open cover of CP k to which we will apply Mayer–Vietoris. We set S D .1; 0; : : : ; 0/ S 1 S 2k

1

:

Let f2 .z 1 ; w/ E WD .z 1 jz 1 j 1 ; 0/ W U2 ! S and let i2 be the inclusion of S in U2 . It is then immediate that f2 i2 D IdS . Deﬁne a homotopy F2 from i2 f2 to IdU2 by: 1 F2 .z 1 ; wI E t/ D .z 1 ; t w/.j.z E ; t w/j/ E

1

:

Consequently, U2 and S are homotopy equivalent spaces. Since the maps in question are S 1 equivariant, O2 is homotopy equivalent to a point. Next, let S 2k 1 WD f.z 1 ; w/ E 2 S 2kC1 W z 1 D 0g; f1 .z 1 ; w/ E WD .0; w=j E wj/ E W U1 ! S 2k 1 :

Let i1 be the inclusion of S 2k 1 into U1 . e same argument as that given above shows that these equivariant maps descend to deﬁne homotopy equivalences between O1 and CP k

1

D S 2k

1

=S 1 :

Finally, a similar argument shows that U1 \ U2 is homotopy equivalent to the space f.z 1 ; w/ E W jz 1 j D jwj E D

p1 g 2

D S 1 S 2k

1

5.1. BASIC PROPERTIES OF DE RHAM COHOMOLOGY

25

1

with the diagonal action .; w/ E D .; w/ E . Let T .; w/ E D .; w/ E deﬁne a diﬀeomor1 2k 1 phism of S S . Since T .; w/ E D .; w/ E , T intertwines the diagonal action of S 1 on 1 2k 1 1 S S with the standard action on S and the trivial action on S 2k 1 . Dividing by the action 1 of S shows that O1 \ O2 is homotopy equivalent to S 2k 1 . erefore, we have p p HdR .O1 / D HdR .CP k

1

/;

p p HdR .O2 / D HdR .pt/;

p p HdR .O1 \ O2 / D HdR .S 2k

1

/:

Consequently, the Mayer–Vietoris sequence becomes: p 1 p p p p HdR .O1 \ O2 / ! HdR .CP k / ! HdR .O1 / ˚ HdR .O2 / ! HdR .O1 \ O2 /

jo

p 1 HdR .S 2k 1 /

p HdR .CP k 1 /

jo ˚

p HdR .pt/

jo

p HdR .S 2k 1 / :

We examine the beginning of the sequence and take p D 0. e spaces in question are path 0 1 connected so HdR ./ D R. Furthermore, by induction HdR .CP k 1 / D 0 so the sequence is: 1 0 ! R ! R ˚ R ! R ! HdR .CP k / ! 0 : q q 1 It now follows that HdR .CP k / D f0g. If 1 q < 2k 1, then HdR .pt/ D HdR .S 2k 1 / D 0 and p p k k 1 the Mayer–Vietoris sequence becomes: 0 ! HdR .CP / ! HdR .CP / ! 0 and, by induction, ( ) 0 if 1 p < 2k 1 is odd p HdR .CP k / D : R if 1 p < 2k 1 is even p Finally, we examine the top of the sequence using the fact that HdR .CP k p D 2k 1: 2k 0 ! HdR

1

2k .CP k / ! 0 ! HdR

2k 2k It now follows HdR .CP k / D R and HdR

1

1

.S 2k

1

1

/ D 0 for p D 2k and

2k / ! HdR .CP k / ! 0 :

.CP k / D 0.

5.1.2 SIMPLE COVER. We say that a ﬁnite open cover U D fOi gi2A of a topological space X is a simple cover if for any subset B of the indexing set A, the intersection \i2B Oi is either contractible or empty. eorem 5.7

Let M be a manifold of dimension m.

1. If M is compact, then there exists a ﬁnite simple cover of M . q 2. If M admits a ﬁnite simple cover, then HdR .M / is ﬁnite-dimensional for any q .

26

5. DE RHAM COHOMOLOGY

Proof. Put an auxiliary Riemannian metric g on M . By Lemma 3.14 in Book I, small geodesic balls are geodesically convex. If M is compact, we may cover M by a ﬁnite collection of small geodesic balls. Since the intersection of geodesically convex sets is geodesically convex (and hence contractible) or empty, this shows that M admits ﬁnite simple covers. Let fOi g1i` be a ﬁnite simple cover of M . We proceed by induction on ` to show HdR .M / is ﬁnite-dimensional. If ` D 1, the result follows since M D O1 is contractible and hence homotopy equivalent to a point. If ` > 1, we let M1 WD [i 0g;

O2 WD f.x; y/ 2 M1 W x < 0g; O4 WD f.x; y/ 2 M1 W y < 0g :

en Oi \ Oj is empty for .i; j / D .1; 2/ or .i; j / D .3; 4/ whilst otherwise the intersection is one of the four open quadrants in the plane. Consequently, this is a simple cover of M1 and M1 is a non-compact manifold which admits a ﬁnite simple cover. Note that the circle x 2 C y 2 D 1 is a deformation retract of M1 so HdR .M1 / D HdR .S 1 /. 2. Let M2 WD R2 [n2N f.n; 0/g be the plane with a countable number of punctures located at discrete points. Let ydx C .x n/dy !n WD : .x n/2 C y 2 One veriﬁes that d!n D 0 and that for < 1 ( Z 0 !k D 2 .x n/2 Cy 2 D 2

if k ¤ n if k D n

)

:

We may then use Stokes’ eorem to see that fŒ!n gn2N are linearly independent elements 1 1 of HdR .M2 / and hence HdR .M2 / is not ﬁnite-dimensional. Consequently, M2 does not admit a ﬁnite simple cover.

5.2

CLIFFORD ALGEBRAS

e following algebraic structure was ﬁrst examined by the English mathematician William Kingdon Cliﬀord and by the German mathematician Rudolf Otto Sigismund Lipschitz.

5.2. CLIFFORD ALGEBRAS

W. Cliﬀord (1845–1879)

27

R. Lipschitz (1832–1903)

roughout Section 5.2, let .V; .; // be a positive deﬁnite inner product space of dimension m. If fe 1 ; : : : ; e m g is an orthonormal basis for V , and if I D .i1 ; : : : ; ip / is a collection of strictly increasing indices 1 i1 < < ip r , we deﬁne e I WD e i1 ^ ^ e ip 2 p .V /. Extend the inner product to the entire tensor algebra and in particular to the space of p -forms p .V /. e fe I gjI jDp then form an orthonormal basis for p .V /. If v 2 V , let ext.v/ W ! ! v ^ !

for

! 2 .V /

be exterior multiplication. Let int.v/ be the dual, interior multiplication; this is characterized by the identity: .ext.v/!; / D .!; int.v// : Let c.v/ WD ext.v/ Lemma 5.8

int.v/ be Cliﬀord multiplication.

Adopt the notation established above. en c.v/2 D kvk2 Id and

c.v/c.w/ C c.w/c.v/ D 2.v; w/ Id; ext.v/ int.w/ C int.w/ ext.w/ D 2.v; w/ Id; ext.v/ ext.w/ C ext.w/ ext.v/ D 0; int.v/ int.w/ C int.w/ int.w/ D 0 :

Proof. Let fe 1 ; : : : ; e m g be an orthonormal basis for V . en ( ) e J for J D I [ f1g if i1 > 1 1 I ext.e /e D ; 0 if i1 D 1 ( ) 0 if i1 > 1 1 I int.e /e D : e J for J D I f1g if i1 D 1

is shows that exterior multiplication by e 1 adds the index 1 if possible while interior multiplication by e 1 cancels the index 1 if possible. It is now immediate that c.e1 /2 D Id. Given an arbitrary vector, we can always choose an orthonormal frame so v D kvke 1 and hence, by rescaling, c.v/2 D kvk2 Id : Polarizing this identity and then decomposing Cliﬀord multiplication into its component graded pieces yields the remaining identities.

28

5. DE RHAM COHOMOLOGY

Fix an orientation of V and let dvol be the oriented volume form. If fe 1 ; : : : ; e m g is an oriented orthonormal basis for V , then dvol D e 1 ^ ^ e m . If ff 1 ; : : : ; f m g is an arbitrary oriented basis for V , let ij WD .f i ; f j /. We then have that dvol D det./

1=2

f 1 ^ ^ f m:

e Hodge ? operator is the linear map from p .V / to m !p ^ ?p p D g.!p ; p / dvol

p

.V / characterized by the relation:

!p ; p 2 p .V / :

for all

For example, ?.e 1 ^ ^ e p / D e pC1 ^ ^ e m . is shows ?m

p ?p

D . 1/p.m

p/

:

e Cliﬀord algebra, Clif.V; .; //, is the universal unital algebra generated by V subject to the Cliﬀord commutation relations v?wCw?v D

2.v; w/ Id :

We use Lemma 5.8 to see that Cliﬀord multiplication gives a well-deﬁned algebra morphism c W Clif.V; .; // ! Hom.V /. e map (5.2.a)

c W ! ! c.!/ 1

deﬁnes a linear map from Clif.V; .; // to .V /. e Cliﬀord commutation relations show that the elements fe i1 ? ? e ip g for 1 i1 < < ip m and 0 p m are a linear spanning set for Clif.V; .; // (if p D 0, the empty product is 1). We show that c is a natural (i.e., basis independent) linear isomorphism from Clif.V; .; // to .V / by computing c.e i1 ? ? e ip /1 D e i1 ^ ^ e ip

for

1 i1 < < ip m :

We emphasize that the algebra structure is not preserved by c since, for example, one has c .e 1 ? e 1 / D c . 1/ D 1 while c .e 1 / ^ c .e 1 / D e 1 ^ e 1 D 0. By an abuse of notation, we shall also let the oriented volume form, dvol, be the associated element Clif.V; .; //; c.dvol/ is then a natural endomorphism of the exterior algebra .V /. Lemma 5.9 Let .V; .; // be an oriented positive deﬁnite inner product space of dimension m. en c.dvol/2 D . 1/m.mC1/=2 Id and ?p D . 1/.2m pC1/p=2 c.dvol/ on p .V /.

Proof. Let fe 1 ; : : : ; e m g be an oriented orthonormal basis for V . We establish the ﬁrst identity by computing:

dvol2 D e 1 ? ? e m ? e 1 ? ? e m D . 1/m e 2 ? ? e m ? e 2 ? ? e m D D . 1/mC.m

1/CC1

Id D . 1/m.mC1/=2 Id,

c.dvol/2 D c.dvol2 / D . 1/m.mC1/=2 Id .

5.2. CLIFFORD ALGEBRAS

29

I

We compare ?!p and c.dvol/!p . We let !p D e where jI j D p . By permuting the elements in the basis, it is suﬃcient to consider the special case in which I D f1; : : : ; pg. We use the isomorphism c of Equation (5.2.a) to replace .V / by Clif.V; .; // and complete the proof by computing: c 1 f?.e 1 ^ ^ e p /g D c 1 fe pC1 ^ ^ e m g D e pC1 ? ? e m ,

c 1 fc.dvol/e 1 ^ ^ e p g

D e 1 ? ? e m ? e 1 ? ? e p D . 1/mC.m

D . 1/.2m

pC1/p=2

1/CC.m pC1/ pC1

e

c 1 .e 1 ^ ^ e p / .

? ? em

e following is a technical result that will play an important role in our discussion of the Maurer–Cartan forms subsequently in eorem 6.30. Let U.k/ be the k -dimensional unitary group.

Lemma 5.10 Trf.g 1 dg/m

1

Let m be even. ere exists a map g W S m g is a non-zero constant multiple of dvolS m 1 .

Proof. If m D 2, we could deﬁne g.x 1 ; x 2 / WD x 1 C 1

2

3

4

g.x ; x ; x ; x / WD

x1 C x3 C

p

p

p

1

! U.k/ so the m

1 form

1x 2 and if m D 4, we could deﬁne

1x 2 1x 4

x3 C x1

p

p

1x 4

!

1x 2

and verify the conclusion of the Lemma directly. For general m, however, it is convenient to use Cliﬀord algebras. Our construction is closely related to Bott periodicity and we refer to Atiyah, Bott, and Shapiro [3] for further details; we mention it only to warn the reader that we are entering into deep waters. Let .V; h; i/ be an m-dimensional positive deﬁnite inner product space. We could take V D Rm with the usual inner product, but it is convenient to work in a coordinate free setting for the moment. Let c be Cliﬀord multiplication. en dc is a 1-form valued endomorphism of .V /. Fix an orientation of V and let dvol be the oriented volume form. Deﬁne a 1-form valued endomorphism of .V / by setting .v/ WD c.v/dc.v/. Let v0 be a unit vector in V . Choose an oriented orthonormal frame fei g for V so that v0 D e1 . Let x D x i ei deﬁne the dual coordinates on V . We have x.v0 / D .1; 0; : : : ; 0/; dx 1 .v0 /jS m 1 D 0; dvolV .v0 / D dx 1 ^ ^ dx m ; dvolS m 1 .v0 / D dx 2 ^ ^ dx m ; dc.v0 /jS m 1 D c.e2 /dx 2 C C c.em /dx m ; c.v0 / D e1 ; and .v0 /jS m 1 D c.e1 /c.e2 /dx 2 C C c.e1 /c.em /dx m :

30

5. DE RHAM COHOMOLOGY

Let 2 i; j m. e commutation relations ( c.e1 /c.ei /c.e1 /c.ej / D i

dx ^ dx

imply .v0 /m 1 jS m

1

j

D

D .m

0

dx j ^ dx i

1/Šc.e1 /c.e2 / c.e1 /c.em

c.e1 /c.ei /c.e1 /c.eiC1 / D

Since m

(

if i ¤ j if i D j

c.e1 /c.ej /c.e1 /c.ei / Cc.e1 /c.ej /c.e1 /c.ei /

if i ¤ j if i D j 1/

) )

˝ dvolS m 1 . If 2 i < m, then

c.e1 /c.e1 /c.ei /c.eiC1 / D c.ei /c.eiC1 / :

1 is odd, we may use this relation to show .v0 /m

1

jS m

1

D .m D .m

1/Šc.e1 /c.e2 / c.em / ˝ dvolS m 1 .v0 / 1/Šc.dvolV .v0 // ˝ dvolS m 1 .v0 /

is independent of the particular unit vector v0 which was chosen. e parity of m mod 4 now enters as ( ) 1 if m 0 mod 4 2 c.dvol/ D : 1 if m 2 mod 4 p 1. Consequently, we may decompose If m 0 mod 4, let D C1; if m 2 mod 4, let D .V / ˝R C D C ˚ into the eigenspaces of c.dvol/ where ˙ WD f! 2 .V / ˝R C W c.dvol/! D ˙!g :

Let v 2 V . Since m is even, c.v/ anti-commutes with c.dvol/ and interchanges C with . erefore, we may decompose c.v/ D cC .v/ C c .v/

where c˙ .v/ W ˙ ! ;

and .v/ D C .v/ C .v/ :

Note that ˙ .v/ WD c .v/dc˙ .v/ belongs to the vector space Hom.˙ ; ˙ / ˝ 1 .V /. e computation performed above then yields Trf˙ .v/m

1

on ˙ .V /g D ˙.m

1/Š dimfV ˙ g dvol

if

v 2 Sm

1

:

Cliﬀord multiplication preserves the inner product and hence is unitary. Let T be a ﬁxed unitary map from to C . Let v 2 V be an arbitrary unit vector, and let g.v/ D T c.v/ deﬁne a smooth map from S m 1 to the unitary group U.C /. en 1

g

so Trf.g

1

dg D cC .v/

dg/m

1

1

T

1

T cC .v/ D cC .v/

1

dcC .v/ D

c .v/dcC .v/ D

g is a non-zero constant multiple of dvol as desired.

C .v/

5.3. THE HODGE DECOMPOSITION THEOREM

5.3

31

THE HODGE DECOMPOSITION THEOREM

e material of this section was introduced by the Italian mathematician E. Beltrami, the Scottish mathematician W. Hodge, and the French mathematician P. Laplace.

E. Beltrami (1835–1899)

W. Hodge (1903–1975)

P. Laplace (1749–1827)

Let .M; g/ be a compact Riemannian manifold. We extend g to a positive deﬁnite inner product on tensors of all types. We integrate this inner product with respect to the Riemannian measure j dvol j to deﬁne a global positive deﬁnite inner product on C 1 .p M / and let L2 .p M / be the L2 completion. Let d W C 1 .p M / ! C 1 .pC1 /

and ı W C 1 .pC1 / ! C 1 .p M /

be exterior diﬀerentiation and the L2 adjoint, respectively. e Laplacian is then given by WD d ı C ıd :

is is also often called the Hodge–Beltrami Laplacian. We will express these operators in terms of the Levi–Civita connection in eorem 5.14 presently; we suppress indices to simplify the notation. e Laplacian is a self-adjoint operator. e spectral theory of a self-adjoint operator in ﬁnite dimensions is very simple; it is diagonalizable. e Laplacian is an operator on an inﬁnitedimensional space and the corresponding theory is more complicated and beyond the scope of this book. We refer instead to Gilkey [20] noting that there are many excellent treatments of this subject. If 2 C 1 .p M /, deﬁne the C k -norm by setting: X kkC k WD sup kr ` k.x/ . x2M `k

eorem 5.11 Let p be the Laplacian acting on p -forms on a compact Riemannian manifold .M; g/ of dimension m.

1. ere exists a complete orthonormal basis fn g for L2 .p M / where n 2 C 1 .p M / satisﬁes p n D n n . e eigenvalues (repeated according to multiplicity) form a discrete set tending to inﬁnity and may be ordered so 0 1 2 . Given > 0 and k 0, there exists an integer ` D `.k; m/ and N D N.; p; M; k/ so that if n N , then: 2

nm

2

< n < n m C

and

kn kC k n` :

32

5. DE RHAM COHOMOLOGY

2. Let n ./ WD .; n /L2 be the Fourier coeﬃcients for 2 L2 .p M /. en the coeﬃcients are “rapidly decreasing” (limn!1 nk n ./ D 0 for every k ) if and only if 2 C 1 .p M /. In this P setting, the expansion D n n ./n of as a generalized Fourier series converges uniformly in the C k topology for all k . In the case of the circle, the decomposition of Assertion 2 is the usual decomposition in terms of Fourier series which was ﬁrst introduced by the French mathematician J. Fourier [19] and later made more precise by the German mathematicians P. Dirichlet and G. Riemann.

P. Dirichlet (1805–1859)

J. Fourier (1768–1830)

G. Riemann (1826–1866)

Let E.; p / WD f 2 C 1 .p M / W p D g be the associated eigenspaces. If Spec.p / is the collection of eigenvalues, then we have a complete orthogonal decomposition L2 .p M / D ˚2Spec.p / E.; p / :

e eigenspaces E.; p / are ﬁnite-dimensional representation spaces for the group of isometries of M ; this will play an important role in our proof of the Peter–Weyl eorem (eorem 6.16) subsequently. 5.3.1 SPHERICAL HARMONICS. We do not want to go too deeply into this subject. Nevertheless, it is worth discussing the spectral resolution of the Laplacian on the sphere to illustrate the phenomena involved. Let xE D .x 0 ; : : : ; x m / 2 RmC1 , and let S m be the unit sphere. Let e WD @2x 0 @2x m be the Euclidean Laplacian, and let s be the spherical Laplacian. Let S.m C 1; j / be the vector space of polynomials which are homogeneous of degree j in m C 1 variables and let H.m C 1; j / be the subspace of harmonic polynomials which are homogeneous of degree j in m C 1 variables: S.m C 1; j / WD ff 2 RŒx 0 ; : : : ; x m W f .t x/ E D t j f .x/ E for t 2 Rg; H.m C 1; j / WD ff 2 S.m C 1; j / W e f D 0g :

For example, if r 2 WD kxk E 2 , then r 2 2 S.2; j /, x i 2 H.m C 1; 1/, and x 0 x 1 2 H.m C 1; 2/. eorem 5.12

Let m 1.

1. If j 2 N , then dimfS.m C 1; j /g D

mCj m

.

5.3. THE HODGE DECOMPOSITION THEOREM 2

2. If j 2 N , then S.m C 1; j / D r S.m C 1; j 3. If j 2 N , then dimfH.m C 1; j /g D mCj m

33

2/ ˚ H.m C 1; j /. mCj 2 . m

4. We have that fj.j C m 1/; H.m C 1; j /gj 2N is the discrete spectral resolution of the Laplacian S m on S m , i.e., j WD j.j C m 1/ for j D 0; 1; 2; : : : is an eigenvalue of S m with associated eigenspace being the restriction of the functions in H.m C 1; j / to the sphere, and there are no other eigenvalues. Proof. It is clear that S.m C 1; j / D x mC1 S.m C 1; j from the resulting recursion relations:

1/ ˚ S.m; j /. Assertion 1 follows

dimfS.m C 1; j /g D dimfS.m C 1; j 1/g C dimfS.m; j /g; dimfS.m C 1; 0/g D 1 and dimfS.1; j /g D 1 : We decompose a homogeneous polynomial p 2 S.m C 1; j / in the form X pD p˛ x ˛ for ˛ D .a1 ; : : : ; amC1 / and x ˛ WD x a1 x amC1 : ˛

P

Let P .p/ WD ˛ p˛ @˛ be the associated partial diﬀerential operator. Deﬁne a Euclidean inner product h; i on S.m C 1; j / by setting: X X hp; qi WD P .p/.q/ D p˛ @˛ fqˇ x ˇ g D ˛Šp˛ q˛ : ˛

˛;ˇ

Let RmC1 D @2x 1 @2x mC1 be the Laplacian on RmC1 . Since P .r 2 / D RmC1 , we have the crucial intertwining relationship hp; RmC1 qi D hr 2 p; qi. Multiplication by r 2 is injective. Since coker.r 2 / D kerfRmC1 g, this proves Assertion 2 and Assertion 3. Identify a homogeneous harmonic function with its restriction to S m . Let A be the subspace of C 1 .S m / which is generated additively by the spaces H.m C 1; j /. Since r 2 jS m D 1, Assertion 2 implies: X H.m C 1; / D fS.m C 1; 2j / C S.m C 1; 2j 1/gjS m ; 2j

A D [j fS.m C 1; 2j / C S.m C 1; 2j

1/gjS m :

Since S.m C 1; j / S.m C 1; k/ S.m C 1; j C k/ and since 1 2 S.m C 1; 0/, A is a unital subalgebra of C 1 .S m /. As the coordinate functions x i 2 H.m C 1; 1/, A separates points of S m . Consequently, by the Stone–Weierstrass eorem, A is dense in C 1 .S m / and hence in L2 .S m /. Introduce parameters x D .r; / on RmC1 for r 2 .0; 1/ and 2 S m and express the Laplacian in the form RmC1 D @2r mr 1 @r C r 2 S m . If f 2 H.m C 1; j /, then RmC1 .f / D 0 so S m f ./ D j.j C m

1/f . /

for

f 2 H.m C 1; j / :

34

5. DE RHAM COHOMOLOGY

Since S m is self-adjoint, E.; S m / ? E.; S m / for 6D . Since H.m C 1; / E.j.j C m

1/; S m / ;

the spaces H.m C 1; j / and H.m C 1; k/ are orthogonal in L2 .S m / for j 6D k . e desired result now follows as L2 .S m / D ˚j H.m C 1; j / ˚j E.j.j C m

1/; S m / L2 .S m / .

t u

If m D 1, eorem 5.12 givesprise to the usual Fourier series decomposition of L2 .S 1 /. We complexify. e map ! cos C 1 sin introduces coordinates on S 1 where is the usual p p p periodic parameter; s D @2 . Let n . / D e 1n = 2 . Set z D x 0 C 1x 1 . en n ./ D z n

for

n0

and n ./ D zN n

for

n < 0:

Since holomorphic and anti-holomorphic functions are harmonic, H.2; n/ D spanC fz n ; zN n g. Furthermore, the spectral resolution of the scalar Laplacian on the circle is given by f.2/ 1=2 n ; n2 gn2Z where the constant function 1 spans the kernel of the Laplacian. e deP composition D n n ./n of eorem 5.11 then becomes the usual decomposition of into Fourier series. More generally, one may consider the m-dimensional torus T m WD S 1 S 1 . Set p E nE D .n1 ; : : : ; nm /; E D .1 ; : : : ; m /; n D e 1En to obtain a complete spectral resolution of the scalar Laplacian T m D @21 f.2/

m=2

@2n as

nE ; kE nk2 gnE 2Zm :

5.3.2 THE HODGE DECOMPOSITION THEOREM [32]. e following result of W. Hodge expresses the de Rham cohomology in terms of the kernel of the Laplacian. We caution the reader that the ring structure is not captured by the harmonic forms; the wedge product of two harmonic forms need not be harmonic. e ﬁniteness of the cohomology groups also follows from eorem 5.7 since a compact manifold admits a ﬁnite simple cover.

Let M be a compact Riemannian manifold. Fix p . ere is an L2 orthogonal direct sum decomposition:

eorem 5.13

C 1 .p M / D kerfp g ˚ d fC 1 .p

1

M /g ; ˚ıfC 1 .pC1 M /g ;

where kerfp g D kerfd g \ kerfıg is ﬁnite-dimensional. e map ! ! Œ! deﬁnes a natural isomorp phism between kerfp g and HdR .M /. Proof. We suppress the index p in the interests of notational simpliﬁcation for the remainder of the proof. We use eorem 5.11 to see that the eigenvalues of the Laplacian tend to inﬁnity; this implies that the kernel is ﬁnite-dimensional. If is a smooth diﬀerential form, then we may

5.3. THE HODGE DECOMPOSITION THEOREM

P

35

expand D n n ./n in a generalized Fourier series where the convergence is uniform in the C k topology for any k . Decompose D ˚0 C ˚1 for ˚0 D

X

nWn D0

n ./n

and ˚1 D

X

nWn ¤0

n ./n D

X

n ./n 1 n :

nWn ¤0

erefore, ˚1 2 rangefg D rangefd ıg C rangefıd g. Consequently, C 1 .p M / D kerfg C rangefd ıg C rangefıd g :

(5.3.a)

Let be a diﬀerential form. If d D 0 and ı D 0, then clearly 2 kerfg. Conversely, D 0 implies 0 D .; /L2 D ..d ı C ıd /; /L2 D .ı; ı/L2 C .d; d/L2 :

Consequently, d D ı D 0

so

kerfg D kerfd g \ kerfıg :

Expand D ˚0 C d ı˚2 C ıd˚3 . en: .˚0 ; ıd˚3 /L2 D .d˚0 ; d ˚3 /L2 D 0; .˚0 ; d ı˚2 /L2 D .ı˚0 ; ı˚2 /L2 D 0; .ıd˚3 ; d ı˚2 /L2 D .ııd ˚3 ; ı˚2 /L2 D 0 :

So Equation (5.3.a) is an orthogonal direct sum decomposition with respect to the L2 inner product. If d D 0, then d ıd˚3 D 0 so: 0 D .d ıd˚3 ; d ˚3 /L2 D .ıd ˚3 ; ıd ˚3 /L2 :

So ıd˚3 D 0 and, consequently, D ˚0 C d ı˚2 and Œ D Œ˚0 . If ˚0 D d˚1 and ˚0 2 kerfg, then we may show ˚0 D 0 and complete the proof by computing: 0 D .ı˚0 ; ˚1 /L2 D .˚0 ; d ˚1 /L2 D .˚0 ; ˚0 /L2 .

t u

We apply the results of Lemma 5.8. Let r be the Levi–Civita connection extended to act on tensors of all types. Let [email protected] ! D !Ij and let [email protected] [email protected] ! D !Ij i ; r! D !Ij dx j

and r 2 ! D !Ij i ˝ dx i ˝ dx j :

Let Rij be the curvature operator of the Levi–Civita connection acting on the exterior algebra. We express the operators d and ı in terms of the Levi–Civita connection in Assertion 1, establish the Weitzenböch identity [61] in Assertion 2, and prove the Bochner Vanishing eorem [5] in Assertion 3 of the following result; Assertion 1 played a crucial role in the proof of eorem 4.9.

36

5. DE RHAM COHOMOLOGY

Let .M; g/ be a compact connected Riemannian manifold without boundary. Let fei g be a local orthonormal frame for TM and let fe i g be the associated dual frame ﬁeld for T M . Let ! be a smooth diﬀerential form on M .

eorem 5.14

1. d! D ext.e i /rei ! and ı! D

int.e i /rei ! .

2. ! D g ij !Iij C 21 c.e i /c.e j /Rij ! . 3. If ! is a smooth 1-form, then 1 ! D

Trfr 2 !g C Ric.!/.

1 4. If Ric 0 and if Ric > 0 at a point of M , then HdR .M / D 0.

Proof. Let A WD ext ır be a natural ﬁrst order operator from C 1 .p M / to C 1 .pC1 M /. Relative to any system of local coordinates, we have A.fI dx I / D dx i ^ [email protected] .fI dx I / D dx i ^ [email protected] i fI /dx I C dx i ^ fI [email protected] dx I D d.fI dx I / C E1

where E1 .fI dx I / WD dx i ^ fI [email protected] dx I :

Since E1 D A d , E1 is a natural 0th order partial diﬀerential operator (i.e., an endomorphism) which is linear in the Christoﬀel symbols. Fix a point P of M . By Lemma 3.13 in Book I, we can choose a system of local coordinates so the ﬁrst derivatives of the metric vanish at P . erefore, the Christoﬀel symbols vanish at P and E1 vanishes at P . Since P was arbitrary, E1 vanishes identically so d D ext ır . is shows that dually n o ı D ext.dx i /[email protected] D int.dx i /[email protected] C E2 D int ır C E2 ; where E2 WD ı int ır is a natural 0th order partial diﬀerential operator which is linear in the ﬁrst derivatives of the metric. e same argument shows E2 D 0 which establishes Assertion 1. We use Assertion 1 to see d C ı D c ı r and, consequently, D .d C ı/2 ! D c.dx i /[email protected] c.dx j /[email protected] ! D c.dx i /c.dx j /!Ij i C E3 .

where again E3 is an invariantly deﬁned 0th order operator which is linear in the 1-jets of the metric; hence again E3 D 0. We now establish Assertion 2 by computing: .d C ı/2 !

D D D

1 .c.dx i /c.dx j /!Ij i C c.dx j /c.dx i /!Iij / 2 1 .c.dx i /c.dx j / C c.dx j /c.dx i //!Ij i C 12 c.dx j /c.dx i /.!Iij 2 g ij !Iij C 12 c.dx j /c.dx i /[email protected] j ; @x i /! :

!Ij i /

We specialize to the case p D 1 and apply Assertion 2. Let fei g be a local orthonormal frame for the tangent bundle and let fe i g be the dual frame. We may then lower and raise indices easily. We have R.ei ; ej /e k D Rij k ` e ` . Consequently, c.e i /c.e j /R.ei ; ej /e k D Rij k ` c.e i /c.e j /c.e ` /1 :

(5.3.b)

5.3. THE HODGE DECOMPOSITION THEOREM

37

Now if the indices fi; j; `g are distinct, then c.e i /c.e j /c.e ` / D c.e j /c.e ` /c.e i / D c.e ` /c.e i /c.e j / :

We have Rij k ` D Rij ` k . e Bianchi identity yields Rij ` k C Rj `i k C R`ij k D 0. Consequently, we may assume that the indices fi; j; `g are not distinct in Equation (5.3.b). We cannot have i D j as Rij ` k C Rj i ` k D 0. Consequently, either i D ` or j D `; this yields the same thing. We suppose i D ` ¤ j . en c.e i /c.e j /c.e ` / D c.e j /. We may derive Assertion 3 from Assertion 2 by computing: 1 c.e i /c.e j /R.ei ; ej /e k 2

Suppose

D Rij k i c.e j /1 D j k e j :

0. en

. 1 !; !/L2

D . !Ii i ; !/L2 C .!; !/L2 D .r!; r!/L2 C .!; !/L2

.!; !/L2

0:

Consequently, if ! is a smooth 1-form with 1 ! D 0, then r! D 0 so ! is parallel and hence k!k is constant. But since > 0 at some point, we have .!; !/.P / D 0 so !.P / D 0. Since k!k is constant, this implies ! vanishes identically which proves Assertion 4. 5.3.3 POINCARÉ DUALITY. e following result is due to the French mathematician J. Poincaré in the topological setting.

Jules Henri Poincaré (1854–1912) Let ? be the Hodge operator and let c.dvol/ be Cliﬀord multiplication by the volume form as discussed in Section 5.2. We shall apply Lemma 5.9 and use the Hodge Decomposition eorem to establish Poincaré duality [53]. If V and W are ﬁnite-dimensional vector spaces, then we say that a map f from V W to R is a perfect pairing if f is bilinear, if given any v in V there exists w in W so that f .v; w/ ¤ 0, and if given any w in W there exists v in V so that f .v; w/ ¤ 0. Equivalently, f exhibits V as the dual of W and W as the dual of V . Let Z I .!p ; m p / WD !p ^ m p for !p 2 C 1 . p M / and m p 2 C 1 . m p M / : (5.3.c) M

38

5. DE RHAM COHOMOLOGY

Let M be a compact connected oriented Riemannian manifold without boundary.

eorem 5.15

1. e Hodge operator ? deﬁnes an isomorphism from kerfp g to kerfm

p

g.

p m p 2. e map I deﬁnes a perfect pairing from HdR .M / HdR .M / to R. p m p 3. dimfHdR .M /g D dimfHdR .M /g. m 4. HdR .M / D Œdvol R is generated by the volume form. R m 5. e map Œ!m ! M !m is an isomorphism from HdR .M / to R.

Proof. We use Stokes’ eorem to show the pairing of Equation (5.3.c) extends to cohomology. Suppose d!p D 0 and dm p D 0. If !p D d p 1 , then d. Z M

p 1

^ m

!p ^ m

D !p ^ m p C . 1/p 1 Z Z D d. p 1 ^ m p / D

p/

p

M

p 1

@M

^ dm p 1

p

^ m

D !p ^ m p

p;

D 0:

Consequently, the pairing extends to cohomology in the ﬁrst factor. We use the identity I .!p ; m p / D . 1/p.m p/ I .m p ; !p / to see that the pairing also extends to cohomology in the second factor. By eorem 5.14, d C ı D c.dx i /[email protected] . Since dvol D . 1/m 1 dvol in Clif.T M / for any 2 T M , c./c.dvol/ D . 1/m 1 c.dvol/c./. Consequently, c.dx i /[email protected] c.dvol/ D . 1/m

1

c.dvol/c.dx i /[email protected] C E ;

where E is linear in the 1-jets of the metric and invariantly deﬁned. Consequently, it vanishes since at any given point P 2 M we can always choose the coordinate system so that the ﬁrst derivatives of the metric vanish at P by Lemma 3.13 in Book I. is proves that: c.dvol/.d C ı/ D . 1/m

1

.d C ı/c.dvol/ :

erefore, c.dvol/ D c.dvol/ so c.dvol/ W kerfp g ! kerfm p g. Since c.dvol/2 D ˙ Id and since c.dvol/ D ˙?, Assertion 1 follows. Let 0 ¤ !p 2 kerfp g. We show that I is a perfect pairing by noting: Z Z I .!p ; ?!p / D !p ^ ?!p D g.!p ; !p /j dvol j D .!p ; !p /L2 ¤ 0 : M

M

0 Assertion 3 follows from Assertion 1. Since M is connected, HdR .M / D R Œ1. Assertion 4 now follows from Assertion 1 since ?1 D dvol. Assertion 5 follows since I .Œ1; / is an isomorphism m from HdR .M / to R.

5.3. THE HODGE DECOMPOSITION THEOREM

39

5.3.4 THE KÜNNETH FORMULA [38]. is result together with Poincaré duality is an extremely powerful tool in examining the ring structure of de Rham cohomology. e following result was established by the German mathematician Künneth.

H. Künneth (1892–1975) eorem 5.16 Let M and N be compact manifolds of dimensions m and n, respectively, and let 1 and 2 be projection on the ﬁrst and second factors, respectively. en 1 ^ 2 is an isomorphism from HdR M ˝ HdR N to HdR .M N /.

Proof. Let gM and gN be Riemannian metrics on M and on N , respectively. We take the product metric g WD gM C gN on M N . Let xE D .x 1 ; : : : ; x m /

and yE D .y 1 ; : : : ; y n /

be local coordinates on M and on N , respectively. We use eorem 5.14 to compute: .d C ı/M N D cM N .dx i /[email protected]N C cM N .dy a /[email protected] aN : x

e Christoﬀel symbols decouple so [email protected]N D [email protected] and [email protected]N D [email protected] a . We have x

y

y

x

c.dx i /c.dy a / C c.dy a /c.dx i / D 2g.dx i ; dy a / D 0; [email protected] c.dy a / D c.dy a /[email protected] ; and [email protected] a c.dx i / D c.dx i /[email protected] a : x

x

erefore, we may compute: 2 D .d C ı/M N D c.dx i /[email protected] c.dx j /[email protected] C c.dx i /[email protected] c.dy a /[email protected] a

M N

x

x

x

Cc.dy a /[email protected] a c.dx i /[email protected] C c.dy a /[email protected] a c.dy b /[email protected] b x

y

D M C N Cc.dx i /c.dy a /[email protected] [email protected] a C c.dy a /c.dx i /[email protected] a [email protected] : x

Since [email protected] i ; @y a / D 0, M N

[email protected] a [email protected] y x

D

[email protected] [email protected] a . y x

x

Consequently,

D M C N C .c.dx i /c.dy a / C c.dy a /c.dx i //[email protected] [email protected] a x D M C N :

40

5. DE RHAM COHOMOLOGY

p M M N Let fp; ; p; g be a complete spectral resolution of M and f q; ; N q; g be a complete spectral q resolution of N . Suppose ﬁrst p D q D 0. Let A be the subalgebra of C 1 .M N / generated by C 1 .M / and C 1 .N /. By the Stone–Weierstrass eorem, A is dense in

C 1 .M N / : M N M Since spanf0; 0; g is dense in A, f0; 2 0 L . .M N //. A similar argument shows M fp; ^

N 0; g

is a complete orthonormal basis for

N q; gpCqDr; 1 C = C CWt 2R : > A > > ; 1

is is immersed but the immersion is not proper as G is not a closed subgroup of GL.4; R/; the topology of G as a Lie group is not the topology induced by the inclusion of G in GL.4; R/. e closure of G in GL.4; R/ is a 2-dimensional torus parametrized by 80 cos./ sin./ ˆ ˆ ˆ

> > =

C C C W ; 2 Œ0; 2 : > A > > ;

e Lie group G winds around in GN with irrational slope and is a dense subgroup.

6.3

THE EXPONENTIAL FUNCTION OF A MATRIX GROUP

Of particular interest in our investigation are matrix groups, i.e., groups which are subgroups of Hom.V / for some ﬁnite-dimensional vector space over F 2 fR; C; Hg. Matters are much more concrete in that setting and a lot of extra formalism can be avoided. By eorem 6.6, the local geometry of a Lie group always arises in this way. In the setting of matrix groups, the exponential map is given by the usual exponential map. We proceed as follows. Let .V; j j/ be a normed vector space. We assume the ﬁeld in question is R to simplify the discussion. Let kAk WD maxjvjD1 jvj be the operator norm deﬁned in Equation (1.2.f ) of Book I. If A; B 2 Hom.V /, then kABk kAk kBk

and kA C Bk kAk C kBk :

56

6. LIE GROUPS

We wish to deﬁne e A D k > j > 2kAk, then ksk .A/

P

n0

An =nŠ. Let sj .A/ D Id CA C C

1 j A jŠ

be the partial sum. If

1 Aj C1 C C sj .A/k D k .j C1/Š

1 k A k kŠ 1 1 kAkj C1 C C kŠ kAkk .j C1/Š 1 1 1 kAkj C1 f1 C j C2 kAk C .j C2/.j kAk2 .j C1/Š C3/ 1 kAkj C1 .1 C 12 C 14 C / .j C1/Š 1 2 .j C1/Š kAkj C1 :

C g

is tends uniformly to zero as j ! 1 and hence, by the Cauchy criteria, we may deﬁne: e A D lim sn .A/ D n!1

Lemma 6.7

1 X An : nŠ nD0

(6.3.a)

Adopt the notation established above.

1. If AB D BA, then e A e B D e ACB . 2. e tA e sA D e .t Cs/A . 3. We may identify TId .GL.V // D Hom.V /. Under this identiﬁcation, expg .A/ D e A . 4. e map A ! e A is a smooth map from Hom.V / to GL.V / which is a local diﬀeomorphism from .Hom.V /; 0/ to .GL.V /; Id/; denote the inverse by log. 5. We have det.e A / D e TrfAg for any A 2 Hom.V /. 6. If kAk < 1, then log.Id CA/ D A

1 2 A 2

C 31 A3 C .

Proof. All the series in question converge absolutely and, consequently, the terms can be rearranged to suit. If A and B commute, we establish Assertion 1 by using the Binomial eorem: .A C B/n D eAeB D

X iCj Dn

nŠ i j AB ; iŠj Š

1 X Ai B j X X .A C B/n 1 X nŠ i j D e ACB : D AB D iŠ j Š nŠ iŠj nŠ Š n nD0 i;j

iCj Dn

Assertion 2 now follows from Assertion 1. Since e A e A D e 0 D Id, e A 2 GL.V /. Let A belong to Hom.V / D TId .GL.V //. en .Lg / A D gA. Let X be the left-invariant vector ﬁeld agreeing with A at the identity; X.g/ D gA for g 2 GL.V /. Let ˚.g; A; t / WD ge tA . By Equation (6.3.a), @ t ˚.g; A; t / D ge tA A D X.˚.g; A; t // :

6.3. THE EXPONENTIAL FUNCTION OF A MATRIX GROUP g

57

A

Consequently, ˚.g; A; t/ gives the ﬂow for X and exp .A/ D ˚.Id; A; 1/ D e which establishes Assertion 3. e map expg is a smooth map from TId GL.V / D Hom.V / to GL.V / by the Fundamental eorem of Ordinary Diﬀerential Equations. erefore, A ! e A is a smooth map; this shows Assertion 4. By replacing V by V ˝R C , we may assume that V is complex in proving Assertion 5 and Assertion 6. By choosing a basis for V , we may assume that V D C m . Suppose ﬁrst that A D diag.1 ; : : : ; m / is diagonal. en e A D diag.e 1 ; : : : ; e m / and Assertion 5 holds. Since the determinant and the trace are independent of the basis chosen, Assertion 5 continues to hold if A is diagonalizable. Since the determinant, trace, and exponential function are continuous, Assertion 5 continues to hold if A can be uniformly approximated by diagonalizable matrices. We may use Jordan normal form to see that the diagonalizable matrices are dense in Mn .C/; Assertion 5 now follows in complete generality. We now prove Assertion 6. e power series in Assertion 6 is the usual power series for the log function. If A is diagonalizable, then the argument given to establish Assertion 5 shows there exists ı > 0 so that e log.Id CA/ D Id CA for 0 < ı . If kAk < 1, then both sides of this equation are well-deﬁned for 0 1 and real analytic in . It now follows using analytic continuation that e log.Id CA/ D Id CA for 2 Œ0; 1 and, consequently, that e log.Id CA/ D Id CA if kAk < 1 and if A is diagonalizable. Again, since the diagonalizable matrices are dense in Hom.V / if V is complex, we conclude this identity holds in complete generality. is shows Assertion 6. u t 6.3.1 COMMUTATOR OF FLOWS. We can relate the commutator of the ﬂows of two vector ﬁelds and their Lie bracket (see also Spivak [57], Ch. 5, Vol. 1).

Let X and Y be smooth vector ﬁelds on a smooth manifold M . Let ˚ tX and ˚ tY be the associated ﬂows. Lemma 6.8

1. Let X.x/ E D a1 .x/@ E x 1 C C am .x/@ E x m D a.x/ E in a system of local coordinates. en X ˚ t D xE C ta.x/ E C 12 t 2 da.x/ E a.x/ E C O.t 3 / . Y X 2. Let .t I P / WD ˚ Ypt .˚ Xpt .˚p .˚p .P ////. en is diﬀerentiable at t D 0 and t t

.P P / tD0 D ŒX; Y .P / .

Proof. Expand the ﬂows in a Taylor series about t D 0:

E D xE C ta1 .x/ E C 12 t 2 a2 .x/ ˚ tX .x/ E C O.t 3 /;

E C O.t 3 / : ˚ tY .x/ E D xE C t b1 .x/ E C 21 t 2 b2 .x/

Note that f .xE C x/ E D f .x/ E C df .x/ E xE C O.kxk E 2 /. We expand: ˚ t .x/ E WD ˚ Yt ˚ Xt ˚ tY .˚ tX .x// E ˚ X Y Y D ˚ t ˚ t ˚ t .xE C ta1 .x/ E C 21 t 2 a2 .x// E C O.t 3 / ˚ E C tb1 .xE C ta1 .x// E C 21 t 2 b2 .x/ E C O.t 3 / : D ˚ Yt ˚ Xt xE C ta1 .x/ E C 12 t 2 a2 .x/

58

6. LIE GROUPS

Since tb1 .xE C ta1 .x// E D tb1 .x/ E C t 2 db1 .x/ E a1 .x/ E C O.t 3 /, we see t .x/ E D ˚ Yt f˚ Xt ŒxE C t a1 .x/ E C tb1 .x/ E 2 1 2 E C 12 t 2 b2 .x/g E C O.t 3 / : Ct db1 .x/ E a1 .x/ E C 2 t a2 .x/

We continue the expansion: E C 12 t 2 b2 .x/ E t .x/ E D ˚ Yt fxE C ta1 .x/ E C t b1 .x/ E C t 2 db1 .x/ E a1 .x/ E C 21 t 2 a2 .x/ 1 2 3 E C O.t / : ta1 .xE C ta1 .x/ E C t b1 .x// E C 2 t a2 .x/g

Since ta1 .xE C ta1 .x/ E C tb1 .x// E D ta1 .x/ E

t 2 da1 .x/ E a1 .x/ E

t 2 da1 .x/ E b1 .x// E C O.t 3 /,

t .x/ E D ˚ Yt fxE C tb1 .x/ E C t 2 db1 .x/ E a1 .x/ E C t 2 a2 .x/ E C 21 t 2 b2 .x/ E 2 2 3 t da1 .x/ E a1 .x/ E t da1 .x/ E b1 .x//g E C O.t / :

Finally, we expand the action of ˚ Yt to compute t .x/ E D xE C tb1 .x/ E 2 Ct db1 .x/ E a1 .x/ E C t 2 a2 .x/ E C 12 t 2 b2 .x/ E t 2 da1 .x/ E a1 .x/ E 2 1 2 E C O.t 3 / : t da1 .x/ E b1 .x/ E tb1 .xE C t b1 .x// E C 2 t b2 .x/

Again, we expand tb1 .xE C b1 .x/t E / D t b1 .x/ E

t 2 db1 .x/ E b1 .x/ E C O.t 3 / to conclude:

t .x/ E D xE C tb1 .x/ E C t 2 db1 .x/ E a1 .x/ E C t 2 a2 .x/ E C 21 t 2 b2 .x/ t 2 da1 .x/ E a1 .x/ E E 1 2 2 2 3 t da1 .x/ E b1 .x// E t b1 .x/ E t db1 .x/ E b1 .x/ E C 2 t b2 .x/ E C O.t / :

e coeﬃcient of t is zero so this simpliﬁes to become: t .x/ E D xE C t 2 fdb1 .x/ E a1 .x/ E da1 .x/ E b1 .x/g E 2 2 2 Ct a2 .x/ E t da1 .x/ E a1 .x/ E C t b2 .x/ E

t 2 db1 .x/ E b1 .x/ E C O.t 3 / :

(6.3.b)

Let X.x/ E D a.x/ E and Y.x/ E D b.x/ E . e deﬁning relation @ t ˚ tX .x/ E D X.˚ t .x// E yields: a1 .x/ E C ta2 .x/ E C O.t 2 / D a.xE C t a1 .x// E C O.t 2 / D a.x/ E C t da.x/ E a1 .x/ E C O.t 2 / :

Consequently, a1 .x/ E D a.x/ E and a2 .x/ E D da.x/ E a.x/ E ; Assertion 1 now follows. Similarly we have b1 .x/ E D b.x/ E and b2 .x/ E D db1 .x/ E b1 .x/ E . erefore, we may rewrite Equation (6.3.b) as: t .x/ E D xE C t 2 db.x/ E a.x/ E

t 2 da.x/ E b.x/ E C O.t 3 / :

(6.3.c)

Let a D .a1 ; : : : ; am / D aip ei and let b D .b 1 ; : : : ; b m / D b j ej where fei g is the standard basis for Rm . We replace ‘t ’ by ‘ t ’ to complete the proof by rewriting Equation (6.3.c) in the form t u pt .x/ E D xE C tei .aj @x j b i b j @x j ai / C O.t 3=2 / D xE C t ŒX; Y C O.t 3=2 /.

6.3. THE EXPONENTIAL FUNCTION OF A MATRIX GROUP

59

6.3.2 LEFT-INVARIANT VECTOR FIELDS IN MATRIX GROUPS. If A belongs to Hom.V /, let XAL (resp. XAR ) be the left-invariant (resp. right-invariant) vector ﬁeld which agrees with A at the identity. e following result relates the bracket on the Lie algebra of GL.V / with the ordinary matrix bracket and illustrates why we have chosen to work with left-invariant rather than with right-invariant vector ﬁelds. L R Let A; B 2 Hom.V /. en ŒXAL ; XBL D XŒA;B and ŒXAR ; XBR D XŒA;B .

Lemma 6.9

L XA

Proof. Let ˚ t

L XB

and ˚ t

be the ﬂows. We use Lemma 6.7 and Lemma 6.8 to see: L L XA XL XL XB p ˚pB ˚pA .x/ ˚ E ŒXAL ; XBL .x/ E D @ t j t D0 ˚ p t t t t ˚ p p p p D @ t j tD0 xE expg . tA/ expg . tB/ expg . tA/ expg . tB/ L D @ t j tD0 fxE Œ1 C .AB BA/t C O.t 3=2 /g D xE ŒA; B D XŒA;B .x/ E , R R XA XR XR XB p ˚pB ˚pA .x/ ˚ ŒXAR ; XBR .x/ E D @ t j tD0 ˚ p E t t t t

˚ p p p p D @ t j tD0 expg . tB/ expg . tA/ expg . tB/ expg . tA/ xE

D @ t j tD0 fŒ1 C .BA

AB/t C O.t 3=2 / xg E D

ŒA; B xE D

R XŒA;B .x/ E .

t u

6.3.3 CLOSED SUBGROUPS OF LIE GROUPS. Let G be a Lie group. A submanifold H of G is said to be a Lie subgroup if H is also a subgroup of G . Note that H is necessarily closed in this setting. e converse is given by the following result. eorem 6.10

A closed subgroup H of a Lie group G is a Lie subgroup.

Proof. We follow the discussion in Hall [26] (pages 75–77). We shall assume that the ambient group is GL.V / as this simpliﬁes the computations; the general case can be proved using exactly the same arguments although with a bit more technical fuss; one can also use eorem 6.6 in this regard. To simplify the notation, we let exp WD expg be given by Equation (6.3.a) and let log WD logg be the local inverse throughout the proof. We set h WD fA 2 Mm .R/ W exp.tA/ 2 H for all tg :

Step 1. We ﬁrst show h is a linear subspace of Hom.V /. If A 2 h, then sA 2 h for all s 2 R since exp.t.sA// D exp..ts/A/. Let A; B 2 h. We compute:

log.exp.tA/ exp.tB// D log..1 C tA/.1 C tB/ C O.t 2 //, log.1 C t.A C B/ C O.t 2 // D t .A C B/ C O.t 2 /, n log exp nt A exp nt B D n log exp nt A exp nt B , 2 2 n nt .A C B/ C O nt 2 D t.A C B/ C O tn , n exp.t.A C B// D lim exp nt A exp nt B . n!1

60

6. LIE GROUPS

Consequently, exp.t.A C B// is the limit of elements of H and belongs to H as H is closed. is shows that A C B 2 h. Step 2. We show next that Œh; h h. Let A; B 2 h. We have ˚ log exp.tA/ exp.tB/ exp. tA/ exp. tB/ D logf.1 C tA C

2 t2 2 A /.1 C tB C t2 B 2 /.1 2 t2 2 B / C O.t 3 /g 2 2 3

tA C

t2 2 A / 2

.1 tB C D logf1 C 0 t C .AB

BA/t C O.t /g D t 2 ŒA; B C O.t 3 /, n2 t 2 ŒA; B D lim log exp. nt A/ exp. nt B/ exp. nt A/ exp. nt B/ , n!1

exp t 2 ŒA; B D lim exp n!1

t A n

exp

t B n

exp

t A n

exp

t B n

n2

.

erefore, exp.t 2 ŒA; B/ 2 H for t 0. Similarly, exp. t 2 ŒA; B/ D exp.t 2 ŒB; A/ 2 H for any t 0. erefore, ŒA; B 2 h. Step 3. We now show that there exists " > 0 so that exp is a local homeomorphism from B" .0/ \ h to a neighborhood of Id in H . Let h? be the complementary subspace. e map .A; A? / ! exp.A/ exp.A? /

is a local diﬀeomorphism from a neighborhood of 0 in Mm .R/ to a neighborhood of Id in GL.m; R/. We suppose the theorem fails and argue for a contradiction. en there exists a sequence of points hn ! Id with hn 2 H and hn … exp.h/. Express hn D exp.An / exp.A? n/:

? ? Taking hQ n WD exp. An /hn D exp.A? n /, we see exp.An / 2 H and An ! 0. Passing to a sub? ? ? Q? sequence, we may assume An D an AQn where AQn are unit vectors with AQ? n ! A and where an ! 0. Let Œ be the greatest integer function and let jn WD Œ atn . en jn 2 H ) exp jn an AQ? ) exp an AQ? exp an AQ? n n 2 H, n 2H Q? exp.t AQ? / D exp. lim jn an AQ? n / D lim exp jn an An . n!1

Q?

n!1

Q?

is shows A 2 h. But A 62 h. us H is a smooth submanifold of GL.m; R/.

6.4

t u

THE CLASSICAL GROUPS

We now use eorem 6.10 to discuss the classical Lie groups (the orthogonal groups, the unitary groups, and the symplectic groups). We refer to Hall [26] and Helgason [30, 31] for further details. Our ambient Lie group will be GL.V / where V is a real, complex, or quaternion vector space with associated Lie algebra glF .V / WD Hom.V / with (by Lemma 6.9) the usual commutator bracket. ese groups will all be closed subgroups of the general group and hence Lie subgroups.

6.4. THE CLASSICAL GROUPS

61

6.4.1 THE GENERAL LINEAR GROUP. We continue the discussion of Section 1.2.4 of Book I. Let F 2 fR; C; Hg; GL.V / is a Lie group with Lie algebra glF .V / D Hom.V /. 6.4.2 THE SPECIAL LINEAR GROUP. SL.V / WD fg 2 GL.V / W det.g/ D 1g over the ﬁeld F D R or F D C . It is necessary to assume F ¤ H as the quaternions are non-commutative and hence the determinant is not deﬁned. By Lemma 6.7, det.e A / D e TrfAg ; the associated Lie algebra (see Section 6.2.4) is given by: sl.V / WD fA 2 Hom.V / W TrfAg D 0g :

6.4.3 THE ORTHOGONAL GROUP. Let O.V; h; i/ WD fg 2 GL.V / W g h; i D h; ig where h; i is a non-degenerate real inner product of signature .p; q/ on a real vector space V . Let A be the adjoint with respect to h; i. e discussion of Section 6.2.4 shows that the Lie algebra is o.V; h; i/ WD fA 2 Hom.V / W A C A D 0g :

6.4.4 THE UNITARY GROUP. Let U.V; h; i/ WD fg 2 GL.V / W g h; i D h; ig where h; i is a Hermitian symmetric inner product on a complex vector space of signature .2p; 2q/. If A is the adjoint, then the associated Lie algebra is u.V; h; i/ WD fA 2 Hom.V / W A C A D 0g :

Let SU.V; h; i/ WD U.V; h; i/ \ SL.V / be the special unitary group; the Lie algebra is given by su.V; h; i/ WD fA 2 Hom.V / W A C A D 0 and TrfAg D 0g. 6.4.5 THE SYMPLECTIC GROUP. Let SpR .V; h; i; J / WD fg 2 GLR .V / W g ˝ D ˝g where ˝.x; y/ WD hx; Jyi is the Kähler form of a pseudo-Hermitian inner product space .V; h; i; J /. e associated Lie algebra is spR .V; h; i; J / WD fA 2 HomR .V / W J t AJ D Ag :

6.4.6 SUMMARY. We collect in the following table the basic information about dimension, compactness and connectedness of the classical Lie groups. We take V D Rm or C m and we take a positive deﬁnite orthogonal or unitary metric.

62

6. LIE GROUPS

Group

Dimension

GL.m; R/

SL.m; C/

Connected

Components

7

7

2

7

X

1

1

7

X

1

2

7

X

1

X

X

1

X

X

1

m

GL.m; C/ SL.m; R/

Compact

2

2m m

2

2

2m

2 2

U.m/

m 2

SU.m/

m

O.m/

m.m

1/=2

X

7

2

SO.m/

m.m

1/=2

X

X

1

1

O.m; C/

m.m

1/

7

7

2

SO.m; C/

m.m

1/

7

X

1

7

X

1

Sp.m; R/

m.m C 1/=2

e following result is a useful observation. Lemma 6.11 to SO.m/.

GL.m/ is homotopy equivalent to O.m/ and SL.m; R/ is homotopy equivalent

Proof. Let A 2 GL.m; R/. By applying the Gram–Schmidt process to the rows of A, we may use matrix multiplication to express A D B C where B 2 T .m/ is a lower triangular matrix with positive entries on the diagonal and where C 2 O.m/. We may express B D diag.e 1 ; : : : ; e m / C BQ where BQ is a strictly lower triangular matrix. Let t .B/ D diag.e t1 ; : : : ; e t m / C t BQ deﬁne a deformation retract of T .m/ to the identity. is shows that T .m/ is contractible and, consequently, GL.m; R/ is homotopy equivalent to O.m/. If A 2 SL.m; R/, then 1 D det.B/ det.C /. Since det.B/ > 0, necessarily det.C / D 1 so C 2 SO.m/. erefore, SL.m; R/ is homotopy equivalent to SO.m/. t u

6.5

REPRESENTATIONS OF A COMPACT LIE GROUP

roughout this section we work over the complex ﬁeld; the situation over the reals is very diﬀerent. Let V be a complex vector space and let G be a compact Lie group. A representation of G on V is a smooth group homomorphism W G ! GLC .V /; .V; / is a said to be a G -module. Let g v WD .g/v

and V G WD fv 2 V W g v D v for all g 2 Gg :

We say that V is irreducible if the only G -invariant subspaces of V are f0g and V . If W is a subspace of V which is invariant under G , then .W; jW / is said to be a submodule. If V1 and V2 are G -modules, let HomG .V1 ; V2 / WD fT 2 Hom.V1 ; V2 / W T V1 D V2 T g I

V and W are isomorphic G -modules if there exists T 2 HomG .V; W / so T is bijective.

6.5. REPRESENTATIONS OF A COMPACT LIE GROUP

63

6.5.1 UNITIARIZING THE REPRESENTATION. We say that a smooth measure d on G is bi-invariant if it is invariant under both left and right translation, i.e., if Lg d D Rg d D d

for all

g 2G:

Let G be a compact Lie group. Let .V; / be a G -module. R 1. ere is a unique smooth bi-invariant measure j dvol j on G with G j dvol j.g/ D 1.

Lemma 6.12

2. G admits a bi-invariant Riemannian metric whose volume element is j dvol j. 3. ere exists a Hermitian inner product .; / on V so that W G ! U.V; .; //. Proof. Let fX1 ; : : : ; Xm g be a basis for the Lie algebra g of left-invariant vector ﬁelds on G and let f! 1 ; : : : ; ! m g be the corresponding dual basis for the left-invariant 1-forms. Let ! WD ! 1 ^ ^ ! m :

Clearly, ! is left-invariant and any left-invariant m-form is a constant multiple of ! . Let j!j be the associated measure. Since left and right multiplication commute, Rh j!j D .h/j!j for .h/ 2 RC . Since Rh1 h2 D Rh2 Rh1 , we have .h1 h2 / D .h1 /.h2 /. Consequently, is a representation from G to RC . Since G is compact, rangefg is bounded. Since is a group homomorphism, rangefg is a subgroup of RRC . Consequently, 1 and the measure j!j is biinvariant. By rescaling ! , we may assume G j!j D 1. We set j dvol j WD j!j to prove Assertion 1; the uniqueness is immediate from the construction. Let g0 be an arbitrary positive deﬁnite inner product on g. We extend g0 to a left-invariant metric on T . If we average g0 over the right action of g on g, we obtain a bi-invariant metric. is metric may be rescaled to ensure the associated volume form has total volume 1 and agrees with the measure deﬁned in Assertion 1. is proves Assertion 2. Let h; i be an arbitrary Hermitian inner product on V . We set Z .v; w/ D h.g/v; .g/wij dvol j.g/ : G

We show that .h/ preserves .; / for any h 2 G and establish Assertion 2 by computing: Z ..h/v; .h/w/ D h.g/.h/v; .g/ .h/wij dvol j.g/ D D

G

Z

Z

Z

G

h.gh/v; .gh/wij dvol j.g/ D

G

h.g/v; Q .g/wij Q dvol j.g/ Q D .v; w/ .

G

h .gh/v; .gh/wij dvol j.gh/ t u

64

6. LIE GROUPS

6.5.2 EXAMPLE. Lemma 6.12 can fail if G is non-compact. By Lemma 6.23, the ax C b group does not admit a smooth bi-invariant measure and the canonical representation of the ax C b group on R2 does not admit an invariant orthogonal inner product. As an other example, we can let G D R and deﬁne a representation of R on C 2 by ! 1 x .x/ WD : (6.5.a) 0 1

is operator has non-trivial Jordan normal form for x ¤ 0 and is conjugate to no element of the unitary group. Consequently, V admits no invariant positive deﬁnite inner product. 6.5.3 DECOMPOSING REPRESENTATIONS INTO IRREDUCIBLES. Lemma 6.13 Let j dvol j be the unique smooth bi-invariant measure of total mass 1 on a compact Lie group G . Let V and W be complex G -modules which are equipped with invariant Hermitian

inner products. 1. Let be orthogonal projection on V G . en D

R

G

.g/j dvol j.g/.

2. If V and W are irreducible G -modules, then ( ) 1 if V is isomorphic to W G dimfHom .V; W /g D . 0 if V is not isomorphic to W 3. Let W be a G -submodule of V . en W ? is a G -submodule of V . 4. ere is an orthogonal direct sum decomposition V D W1 ˚ ˚ W` of V into non-trivial irreducible submodules. If W is any irreducible G -module, let n.V; W / be the number of the summands Wi which are isomorphic to W . en n.V; W / D dimfHomG .W; V /g is independent of the particular decomposition chosen. Proof. If v 2 V G is a ﬁxed vector, then .g/v D v for all g in G . erefore, Z Z v D .g/vj dvol j.g/ D vj dvol j.g/ D v : G

G

is shows D Id on V G . We show .v/ 2 V G for any v 2 V by computing: Z Z .h/v D .h/ .g/vj dvol j.g/ D .hg/vj dvol j.g/ ZG ZG D .g/vj dvol j.h 1 g/ D .g/vj dvol j.g/ D v : G

G

6.5. REPRESENTATIONS OF A COMPACT LIE GROUP G

G

65

2

is shows rangefg D V . Since is the identity on V , D . Since h; i is an inner product invariant under the action of G , .g/ D .g/ 1 . We see that is self-adjoint and establish Assertion 1 by computing: Z Z v D .g/vj dvol j.g/ D .g/ 1 vj dvol j.g/ G G Z Z D .g 1 /vj dvol j.g/ D .g/vj dvol j.g 1 / G G Z D .g/vj dvol j.g/ D v : G

Let V be irreducible. Let T 2 HomG .V / and let be a complex eigenvalue of T . Set E WD fv 2 V W T v D vg :

If v 2 E and if g 2 G , then T .g v/ D g T v D g v so E is a non-trivial G -invariant subspace. Since V is irreducible, we conclude E D V and T D Id. Consequently, dimfHomG .V /g D 1; it now follows dimfHomG .V; W /g D 1 if W is isomorphic to V . Conversely, suppose that dimfHomG .V; W /g ¤ 0. Choose T non-zero in HomG .V; W /. Since T is non-zero and T .g v/ D g T v , rangefT g ¤ f0g. Consequently, since W is irreducible, rangefT g D W . Since kerfT g ¤ V and V is irreducible, kerfT g D f0g. Consequently, T provides a G isomorphism from V to W . is proves Assertion 2. Let W be a G -invariant subspace of V . Let w 2 W , let w ? 2 W ? , and let g 2 G . Since 1 .g /w 2 W , ..g 1 /w; w ? / D 0. We show .g/w ? 2 W ? and establish Assertion 3: .w; .g/w ? / D ..g

1

/w; .g

1

/.g/w ? / D . .g

1

/w; w ? / D 0 :

By applying Assertion 3 recursively, we can construct an orthogonal direct sum decomposition V D W1 ˚ ˚ W` of V into irreducible modules Wi . We complete the proof of Assertion 4 by computing: dimfHomG .W; V /g D dimfHomG .W; W1 / ˚ ˚ HomG .W; W` /g

D dimfHomG .W; W1 /g C C dimfHomG .W; W` /g D n.V; W / .

Assertion 2 of Lemma 6.13 fails over R. Let G D S 1 . Let ! cos sin T ./ WD sin cos

u t

deﬁne an action of S 1 on R2 by rotations. Since the only invariant subspaces are R2 and f0g, 1 this representation is irreducible. However, Id 2 HomS .R2 ; R2 / and, since G is Abelian, T . /

66

6. LIE GROUPS 1

1

belongs to HomS .R2 ; R2 / for any . is shows dimfHomS .R2 ; R2 /g 2. What happens, of course, is that T ./ does not have any eigenspaces for 2 .0; / in contrast to the complex setting. Assertion 4 of Lemma 6.13 fails if G is not compact. Let W R ! GLC .2/ be deﬁned by Equation (6.5.a). e invariant subspaces of are f0g, V1 WD C .1; 0/, and C 2 . ere is no complementary invariant subspace to V1 and we cannot decompose C 2 D V1 ˚ V2 as the direct sum of irreducibles; exhibits non-trivial Jordan normal form and is not unitarizable in this instance. 6.5.4 THE ORTHOGONALITY RELATIONS. Let G be a compact Lie group. Let L2 .G/ be deﬁned by the bi-invariant smooth measure of total volume 1 which is given by Lemma 6.12. If V is a complex G -module, let .; / be a positive deﬁnite G -invariant inner product on V given by j Lemma 6.12. Choose an orthonormal basis fei g for V and let .g/ei D V;i .g/ej be the matrix coeﬃcients. Lemma 6.14

Let G be a compact Lie group. Let V and W be irreducible complex G -modules.

j v 1. If V and W are inequivalent, then .W;u ; V;i /L2 .G/ D 0 for all i , j , u, v . j v 2. .V;u ; V;i /L2 .G/ D

1 dimfV g ıui ıvj

for all i , j , u, v .

Proof. We identify Hom.V; W / D V ˝ W to deﬁne a natural G action on Hom.V; W / so that g.T / D W .g/ ı T ı V .g/ ; the natural inner product on Hom.V; W / is then G -invariant as well. Because V .g/ D V .g 1 /, g.T / D gT g 1 . Consequently, if T belongs to HomG .V; W /, then gT D T g . Fix i and v . We use Lemma 6.13 to express: Z j v .T /vi D W;u .g/ Tju V;i .g/ j dvol j.g/ (6.5.b) G R v i v i D Tju G W;u .g/NV;j .g/j dvol j.g/ D Tju .W;u ; V;j /L2 .G/ for all i; v :

If V and W are irreducible and inequivalent, then HomG .V; W / D f0g by Lemma 6.13. Consequently, the orthogonal projection from Hom.V; W / to HomG .V; W / is the zero map. v i v i us, .T /vi D 0 so 0 D Tju .W;u ; V;j /L2 .G/ . As Tju was arbitrary, .W;u ; V;j /L2 .G/ D 0 for all j; u as well. is proves Assertion 1. Let dimfV g D r . If V D W , then HomG .V; V / D C Id by Lemma 6.13. We take a basis fAj u g for Hom.V / by setting .Aj u /vi WD ıji ıuv . We have Aj u W ej ! eu . is is the matrix whose only non-zero entry is in position .u; j /. e fAj u g are an orthonormal basis for Hom.V /. Since P Id D u Auu , k Id k2 D r . Since Aab has a 1 in position .a; b/ and is zero elsewhere, hAab ; Idi is 1 if a D b and 0 otherwise. We use Equation (6.5.b) to establish Assertion 2 by computing: r

1

ıab ıiv D r

1

i v i v ; V;a /L2 .G/ . t u hAab ; Idi Idvi D .Aab /vi D .Aab /ju .V;u ; V;j /L2 .G/ D .V;b

6.5. REPRESENTATIONS OF A COMPACT LIE GROUP

67

6.5.5 THE PETER–WEYL THEOREM. e left regular action .Lg f /.h/ WD f .gh/ makes L2 .G/ into a representation space G ; L2 .G/ is ﬁnite-dimensional if and only if G is a ﬁnite group. Let IrrC .G/ be the set of equivalence classes of irreducible complex modules for the group G . By j Lemma 6.14, fV;i gV 2IrrC .G/ is an orthogonal subset of L2 .G/. Consequently, in particular, all these functions are linearly independent. If V 2 IrrC .G/, let: j j AjV WD span1i dimfg fV;i g L2 .G/ and AV WD span1i;j dimfg fV;i g L2 .G/ .

Lemma 6.15

Let V; W 2 IrrC .G/.

1. dimfAjV g D dimfV g and dimfAV g D dimfV g2 . 2. AjV ? AiV in L2 .G/ for i ¤ j . 3. AV ? AW in L2 .G/ for V not isomorphic to W .

4. Lg AjV D AjV so AjV is a ﬁnite representation space for G . 5. AjV is isomorphic to V as a representation space for G . 6. Let AQ be a ﬁnite-dimensional G -invariant subspace of L2 .G/ which is abstractly isomorphic to V as a representation space. en AQ AV in L2 .G/. j 7. fV;i g2IrrC .G/ is a complete orthogonal basis for L2 .G/.

8. L2 .G/ D ˚2IrrC .G/ ˚1j dimfg Aj . Proof. e ﬁrst three assertions follow from the orthogonality relations of Lemma 6.14. We now establish Assertion 4 and Assertion 5. We have that j j j ` fLg V;i g.h/ D V;i .gh/ D V;i .g/V;` .h/ : j j ` is means we have the functional identity Lg V;i D V;i .g/V;` . Consequently, the space Aj

j is invariant under Lg and the corresponding matrix representation is given by V;i relative to 2 Q the canonical basis. Finally, suppose A is a subspace of L .G/ whose dimension is ﬁnite. Also assume AQ is abstractly isomorphic to V as a representation space. Choose a basis fi for AQ so that Lg f D V; .g/f . Evaluating at the unit of the group shows P .g/f .e/ : f .g/ D .Lg f /.e/ D V; erefore, f is a linear combination of the V; so AQ A. is proves Assertion 6. 2 We use eorem 5.11 to decompose L .G/ D ˚ E./ into the eigenspaces of the scalar Laplacian. Each eigenspace E./ is a representation space for G whose dimension is ﬁnite. We decompose each E./ as the direct sum of irreducible modules. Each irreducible module is a subspace of some A and, consequently, E./ ˚ A . is shows L2 .G/ ˚ A ; the reverse inclusion is trivial. t u

68

6. LIE GROUPS

e following result, called the Peter–Weyl eorem [52], was proved by Hermann Weyl and his student Fritz Peter (1899–1949) and follows from the discussion above.

H. Weyl (1885–1955) j

eorem 6.16 (Peter–Weyl). Let G be a compact Lie group. en fV;i g2IrrC .G/ is a complete orthogonal basis for L2 .G/. Furthermore, L2 .G/ D ˚2IrrC .G/ dimfg as a representation space for G under left multiplication. Remark 6.17 Let G be a compact Lie group which is equipped with a bi-invariant metric for G . Let E be the eigenspaces of the scalar Laplacian. We use eorem 5.11 to decompose decompose L2 .G/ as an orthogonal direct sum in the form L2 .G/ D ˚ E . Each E is a ﬁnitedimensional representation space for G and, consequently, decomposes as the direct sum of irreducibles. erefore, we may assume that the functions ab are all eigenfunctions for the Laplacian. We may decompose a smooth function 2 C 1 .G/ into a generalized Fourier series: X D kab kL22 .; ab /L2 ab : a;b

One must renormalize the Fourier coeﬃcients as the functions ab are an orthogonal but not an orthonormal basis for L2 . is series converges in the C 1 topology. eorem 6.18 If G is a compact Lie group, then there exists a smooth representation which embeds G into the orthogonal group SO.p/ for some p . Consequently, every compact Lie group can be regarded as a closed subgroup of a matrix group. is can fail if G is non-compact. By Lemma 6.11, SL.2; R/ is homotopy equivalent to SO.2/ D S 1 and, therefore, 1 .SL.2; R// D Z. e universal cover SL.2; R/ is an example of a group that is not a matrix group; we will show in Lemma 6.25 that this group admits no faithful ﬁnite-dimensional representation. However, locally G can always be identiﬁed with a matrix group by eorem 6.6.

Proof. Let f1 ; : : : g be an enumeration of IrrC .G/; the collection of isomorphism classes of irreducible representations of G is countable by eorem 6.16. Let ni WD dimfi g. Each i is a group homomorphism from G to GLC .ni / and k WD 1 ˚ ˚ k is a group homomorphism from G to GLC .n1 / GLC .nk / GLR .mk / where mk WD 2.n1 C C nk /. We

6.6. BI-INVARIANT PSEUDO-RIEMANNIAN METRICS

69

b i;a

will show that k is an embedding for k large. Let be the complex-valued coeﬃcient functions of each i where 1 a; b ni . It suﬃces to show that the collection of all the functions b b f 0 and q > 0 so G has four connected components. Lemma 6.22

Let G D O.V; h; i/ where h; i has signature .p; q/. Let m D dimfV g D p C q .

1. K.; / D .m

2/ Trfg has signature .p.p

1/ C q.q

1/; pq/.

2. If m ¤ 4 and if is a bi-invariant symmetric bilinear form on g, then D K for some . erefore, in particular, G does not admit a bi-invariant Riemannian metric if p > 0, q > 0, .p; q/ ¤ .1; 1/, and m ¤ 4.

6.7. THE KILLING FORM

73

Proof. Let g D fA 2 Hom.V / W hAv; wi C hv; Awi D 0g be the Lie algebra of O.V; .; //. Let fe1 ; : : : ; em g be an orthonormal basis for V . Consequently, hei ; ej i D 0 for i ¤ j while there is a choice of signs i D ˙1 so hei ; ei i D i . Set 8 9 if k D j > ˆ < ei = Aij ek D : i j ej if k D i ˆ > : ; 0 otherwise

If fi; j; k; `g are distinct indices, then Aij Ak` D 0 so TrfAij Ak` g D 0. If fi; j; kg are distinct indices, then Aij Aik ek D Aij ei D

i j ej ;

Aij Aik ei D

i k Aij ek D 0;

TrfAij Aik g D 0 :

Finally, Aij Aij ej D Aij ei D i j ej and Aij Aij ei D i j Aij ej D i j ei so TrfAij Aij g D 2i j : is shows that Trfg is a non-degenerate bi-invariant pseudo-Riemannian metric on G of signature .p.p 1/ C q.q 1/; pq/. Let fi; j; kg be distinct indices. e only non-trivial commutators are of the form ŒAij ; Ai k . We compute: .Aij Aik .Aij Aik

Ai k Aij /ek D i j ej D i j Aj k ek ; Ai k Aij /ej D i k ek D i j Aj k ej ;

so ŒAij ; Aik D i j Aj k . If fi; j; k; `g are distinct indices, K.Aij ; Ak` / D 0. We examine: ad.Aij / ad.Aik /Aij D ? ad.Aij /Akj D ?Ai k ; ad.Aij / ad.Aik /Ai n D ? ad.Aij /Ak n D 0; ad.Aij / ad.Aik /Aj k D ? ad.Aij /Aij D 0; ad.Aij / ad.Aik /Ank D ? ad.Aij /Ai n D ?Aj n ; where ? is a ˙ sign not of interest. is shows K.Aij ; Ai k / D 0. Finally, we compute ad.Aij / ad.Aij /Aik D i j ad.Aij /Aj k D ad.Aj i /Aj k D j i Ai k ; ad.Aij / ad.Aij /Aj k D i j ad.Aij / ad.Aj i /Aj k D ad.Aij /Aik D i j Aj k : Consequently, K.; / D 2.m 2/i j D .m 2/ Trfg and Assertion 1 follows. Let be a bi-invariant symmetric bilinear form on g. If m D 2, then g D R and is unique. Suppose m 3. Let fi; j; kg be distinct indices. Suppose j D k . We deﬁne the rotation in G by setting: Tj;k 8 9 if ` D j > ˆ < cos ej C sin ek = Tj;k; e` D : sin ej C cos ek if ` D k ˆ > : ; e` otherwise

74

6. LIE GROUPS

1 We have Tj;k; Aij Tj;k; D cos Aij C sin Ai k . erefore, 1 1 .Aij ; Aij / D .Tj;k; Aij Tj;k; ; Tj;k; Aij Tj;k; /

D cos2 .Aij ; Aij / C 2 cos sin .Aij ; Aik / C sin2 .Ai k ; Aik / :

is equality for all implies .Aij ; Aij / D .Ai k ; Aik / and .Aij ; Ai k / D 0. If j ¤ k we replace cos and sin by cosh and sinh and change a sign to deﬁne the hyperbolic boost 8 ˆ < cosh ej C sinh ek Tj;k; e` D sinh ej C cosh ek ˆ : e`

if ` D j if ` D k otherwise

9 > = > ;

:

e same argument shows .Aij ; Aij / D cosh2 .Aij ; Aij / C 2 cosh sinh .Aij ; Aik / C sinh2 .Ai k ; Aik / :

We now conclude .Aij ; Aij / D .Ai k ; Aik / and .Aij ; Ai k / D 0. Combining these two calculations yields i j .Aij ; Aij / D i k .Aik ; Ai k / and .Aij ; Ai k / D 0. And permuting the indices recursively lets us show: i j .Aij ; Aij / D u v .Auv ; Auv /

and .Aij ; Aik / D 0

for

fi; j; kg

distinct :

is shows is unique if m D 3. For m > 4, it only remains to examine .Aij ; Ak` / for fi; j; k; `g distinct. Let fi; j; k; `; ng be distinct indices. We suppose ` D n as the other case is similar. We compute 1 1 .Aij ; Ak` / D .T`;n; Aij T`;n; ; T`;n; Ak` T`;n; / D cos .Aij ; Ak` / C sin .Aij ; Ak` / :

is identity for all implies .Aij ; Ak` / D 0 and completes the determination of .

t u

We shall show presently in Lemma 6.24 that if m D 4, then the universal cover of SO.4/ is S 3 S 3 and, consequently, so.4/ D so.3/ ˚ so.3/ and the space of bi-invariant symmetric bilinear forms has dimension 2; 4-dimensional geometry is very diﬀerent in this regard. 6.7.2 THE KILLING FORM OF THE AX C B GROUP. e following is a useful example. We consider the set of linear transformations fu;v .x/ D ux C v , for u > 0, which we identify with RC R. en: .fu;v ı fu; Q C .uvQ C v/ D fuu;u : Q vQ /.x/ D uux Q vCv Q

We give R R a group structure by setting .u; v/ ? .u; Q v/ Q D .uu; Q uvQ C v/.

6.7. THE KILLING FORM

Lemma 6.23

75

Let G be the ax C b group.

1. ere is a basis for the Lie algebra g so ŒX L ; Y L D Y L , K.X L ; X L / D 1, K.X L ; Y L / D 0, and K.Y L ; Y L / D 0 .

is group does not admit a bi-invariant measure. 2. If W G ! SO.m/ is a group homomorphism, then W g ! so.m/ is not injective. In particular, the canonical representation of G on R2 does not admit an invariant positive deﬁnite inner product. Proof. We have: .La;b / [email protected] / D [email protected] ; La;b .du/ D adu; .La;b / [email protected] / D [email protected] ; La;b .dv/ D adv; R.a;b/ .u; v/ D .au; v C ub/; .Ra;b / [email protected] / D [email protected] C [email protected] ; Ra;b .du/ D adu; .Ra;b / [email protected] / D @v ; Ra;b .dv/ D dv C bdu : L.a;b/ .u; v/ D .au; av C b/;

Let X L=R (resp. Y L=R ) be the left/right-invariant vector ﬁeld agreeing with @u (resp. @v ) at .1; 0/, and let !XL=R (resp. !YL=R ) be the left/right-invariant 1-form agreeing with du (resp. dv ) at .1; 0/, then X L .u; v/ D [email protected] ; Y L .u; v/ D [email protected] ; !XL D u 1 du; !YL D u 1 dv; ŒX L ; Y L D Y L ; K.X L ; X L / D 1; K.X L ; Y L / D 0; K.Y L ; Y L / D 0; d!XL D 0; d!YL D !XL ^ !YL ; X R .u; v/ D [email protected] C [email protected] ; Y R .u; v/ D @v ; !XR D u 1 du; !YR D dv u 1 du; ŒX R ; Y R D Y R ; K.X R ; X R / D 1; K.X R ; Y R / D 0; K.Y R ; Y R / D 0; d!XR D 0; d!YR D !XR ^ !YR :

e left-invariant volume element can be taken to be given by !XL ^ !YL D u 2 dudv . e rightinvariant volume element can be taken to be !XR ^ !YR D u 1 dudv . ese are diﬀerent; there is no bi-invariant volume form and hence no bi-invariant pseudo-Riemannian metric. e action on 1 .g/ is given by: of Ra;b .u .!XL / D Ra;b Ra;b

.u .!YL / D Ra;b Ra;b 1

1 1 1

du/ D !XL ;

dv/ D .au/

D ba u du C au L L Ra;b .!X ^ !Y / D a 1 !XL ^ !XL :

1

1

.dv C bdu/

dv D ba

1

!XL C a

1

!YL ;

76

6. LIE GROUPS

e action of Ra;b on 1 .g/ deﬁnes a group homomorphism from G to M2 .R/ and exhibits G as a matrix group: ! 1 ba 1 .a; b/ WD ; 0 a 1 ! 1 a 1c 1b C c 1d ..a; b/ .c; d // D .ac; b C ad / D ; 0 a 1c 1 ! ! 1 ba 1 1 dc 1 .a; b/.c; d / D : 0 a 1 0 c 1

We will return to the ax C b group in Section 7.4.3 when we construct a left-invariant metric on this group which is Lorentzian and incomplete. We conclude by remarking that K.X; X/ D 1;

K.X; Y / D 0;

K.Y; Y / D 0

so the Killing form is positive semi-deﬁnite and degenerate in this instance. is proves Assertion 1. eorem 6.21 implies g is not a Lie subalgebra of so.m/ for any m so there is no so is injective. Assertion 2 follows. t u

6.8

THE CLASSICAL GROUPS IN LOW DIMENSIONS

ere are some relationships among the low-dimensional Lie groups which are important. Lemma 6.24

1. SO.3/ D S 3 =Z2 D RP 3 where Z2 D f˙1g. e round metric on S 3 induces a bi-invariant Riemannian metric on SO.3/. 2. SU.2/ D S 3 . e round metric on S 3 induces a bi-invariant Riemannian metric on SU.2/. 3. SO.4/ D .S 3 S 3 /=Z2 where Z2 D f˙1g acts diagonally on S 3 S 3 . ere is a 2parameter family of bi-invariant Riemannian metrics on SO.4/ corresponding to rescaling the round metrics on each factor of S 3 S 3 . Proof. Let H be the quaternions. If g 2 S 3 , let .g/x D gx gN . Since k.g/k2 D gx ggx N gN D gx gg N xN gN D gx xN gN D gkxk2 gN D kxk2 g gN D kxk2 ; .g/ 2 O.4/. Since .g/1 D 1, .g/ restricts to deﬁne a map .g/ W 1? ! 1? . Identify R3 D x 1 i C x 2 j C x 3 k

6.8. THE CLASSICAL GROUPS IN LOW DIMENSIONS 3

77

3

with the purely imaginary quaternions. We then have W S ! O.4/. Since S is connected, is a map from S 3 to SO.3/. Let gi ./ D cos C sin i;

gj . / D cos C sin j;

gk ./ D cos C sin k :

Let fA12 ; A23 ; A31 g be the basis for so.3/ deﬁned above and let fAi D i; Aj D j; Ak D kg be the natural basis for the Lie algebra of S 3 . en .gi .//i D .cos C sin i/i.cos

sin i / D i ,

.gi .//k D .cos C sin i /k.cos

sin i / D

.gi .//j D .cos C sin i/j.cos

.gj .//i D .cos C sin j /i.cos

.gj .//j D .cos C sin j /j.cos

.gj .//k D .cos C sin j /k.cos

.gk .//i D .cos C sin k/i.cos

.gk .//j D .cos C sin k/j.cos .gk .//k D .cos C sin k/k.cos .Ai / D .g P i /.0/ D

sin i / D cos.2 /j C sin.2/k ,

sin.2/j C cos.2 /k ,

sin j / D cos.2/i

sin.2/k ,

sin j / D j ,

sin j / D sin.2/i C cos.2 /k ,

sin k/ D cos.2 /i C sin.2/j , sin k/ D

sin k/ D k ,

sin.2 /i C cos.2/j ,

2A23 , .Aj / D .g P i /.0/ D

2A31 , .Ak / D

2A12 :

is shows that is a Lie group isomorphism from the Lie algebra of S 3 to so.3/ and hence is a diﬀeomorphism from a neighborhood of 1 2 S 3 to a neighborhood of Id in SO.3/. Consequently, is an open map on a neighborhood of 1. Since .gh/ D .g/ .h/ ; is an open map. Since S 3 is compact, is a closed map. Since SO.3/ is connected, is surjective. erefore, is a local diﬀeomorphism from S 3 onto SO.3/ and hence is a covering projection. Suppose .x/ D Id. en xy xN D y for all y 2 S 3 and hence xy D yx for all y 2 S 3 . is implies x is scalar so x D ˙1 and kerfg D ˙1. Consequently,

SO.3/ D S 3 =Z2 is the sphere with antipodal points identiﬁed. is implies that SO.3/ D RP 3 so S 3 has a unique bi-invariant metric which is easily seen to be the round metric. We now study SU.2/. Let ! ! ! ! 1 0 0 1 0 i i 0 e0 D ; e1 D ; e2 D ; e3 D : (6.8.a) 0 1 1 0 i 0 0 i p p Let ˛ WD x 0 C 1e 3 and ˇ WD x 1 C 1e 2 with ˛ ˛N C ˇ ˇN D .x 0 /2 C .x 1 /2 C .x 2 /2 C .x 3 /2 D 1 :

78

6. LIE GROUPS

We decompose g 2 SU.2/ in the form: ! ˛ ˇ gD D x 0 e0 C x 1 e1 C x 2 e2 C x 3 e3 : ˇN ˛N We establish the required identiﬁcation of SU.2/ with S 3 by showing that the quaternion relations are satisﬁed: e12 D e22 D e32 D

Id; e1 e2 D e2 e1 D e3 ; e2 e3 D e3 e2 D e1 ; e3 e1 D e1 e3 D e2 : (6.8.b)

We now examine SO.4/. Let .g1 ; g2 /g WD g1 gg2 for g1 ; g2 ; g 2 S 3 . We compute: k.g1 ; g2 /gk2 D g1 gg2 gN 2 gN gN 1 D 1

and, consequently, deﬁnes a representation from S 3 S 3 to SO.4/. If .g1 ; g2 / D Id, then .g1 ; g2 /1 D 1 and, therefore, g1 gN 2 D 1 so g1 D g2 . e argument given above to study SO.3/ then shows g1 D g2 D ˙1 so W .S 3 S 3 /=f˙.1; 1/g ! SO.4/

is injective. It is smooth. Let be a left-invariant vector ﬁeld on S 3 S 3 . If D 0, then the ﬂow for is trivial so .e t / D Id; this is not possible as is locally injective. is implies is injective. Since dimfS 3 S 3 g D dimfSO.4/g D 6, we conclude is a covering projection so SO.4/ D .S 3 S 3 /=f˙.1; 1/g. t u Let SL.2; R/ be the group of invertible 2 2 invertible real matrices of determinant 1. ere is a close relationship between SL.2; R/ and SU.2/ which we will exploit; this relates the Cliﬀord algebra on a metric of signature .0; 3/ with the Cliﬀord algebra on a metric of signature .2; 1/. Lemma 6.25

1. e Killing form is a bi-invariant Lorentzian metric on SL.2; R/, SL.2; R/ admits a biinvariant volume form, and SL.2; R/ is geodesically complete. 2. No non-trivial representation of SL.2; R/ admits an invariant Euclidean inner product. 3. e exponential map expg is not surjective. 4. 1 .SL.2; R// D Z. 5. Let SL.2; R/ be the universal covering group of SL.2; R/. en no ﬁnite-dimensional representation of SL.2; R/ is faithful so SL.2; R/ is not a matrix group.

6.8. THE CLASSICAL GROUPS IN LOW DIMENSIONS

79

Proof. Let fe0 ; e1 ; e2 ; e3 g be as deﬁned in Equationp (6.8.a); these satisfy p the quaternion relations of Equation (6.8.b). Let f0 D e0 , f1 D e1 , f2 D 1e2 , and f3 D 1e3 . en: ! ! ! ! 1 0 0 1 0 1 1 0 f0 D ; f1 D ; f2 D ; f3 D : 0 1 1 0 1 0 0 1

e fi ’s satisfy the Cliﬀord commutation relations for a Cliﬀord algebra of signature .2; 0/: fi fj C fj fi D 0 for 1 i < j 3; f12 D Id; f22 D f32 D Id; f1 f2 D f3 ; Œf1 ; f2 D 2f3 ; Œf2 ; f3 D 2f1 ; Œf3 ; f1 D 2f2 :

Let e.x/ D x i ei and let f .x/ D y i fi . en: ! x 0 C ix 3 x 1 C ix 2 e.x/ D and f .x/ D x 1 C ix 2 x 0 ix 3

y0 C y3 y1 C y2

y1 C y2 y0 y3

!

Let .x; x/ Q WD x 0 xQ 0 C x 1 xQ 1 C x 2 xQ 2 C x 3 xQ 3 and let hy; yi Q WD y 0 yQ 0 C y 1 yQ 1 y 2 yQ 2 en SU.2/ WD fe.x/ W .x; x/ D 1g and SL.2; R/ WD ff .y/ W hy; yi D 1g :

: y 3 yQ 3 .

We extend e./ to be complex linear and .; / to be complex bi-linear. Let SUC .2/ WD fe.z/ W .z; z/ D 1g GL.2; C/ : en both SU.2/ and SL.2; R/ are Lie subgroups of SUC .2/ and we pass from SU.2/ to SL.2; R/ by setting x 2 D iy 2 and x 3 D iy 3 . We have ad.e1 /e1 D 0; ad.e1 /e2 D 2e3 ; ad.e1 /e3 D 2e2 ; ad.e2 /e1 D 2e3 ; ad.e2 /e2 D 0; ad.e2 /e3 D 2e1 ; ad.e3 /e1 D 2e2 ; ad.e3 /e2 D 2e1 ; ad.e3 /e3 D 0 : It now follows that K.ei ; ej / D 8ıij so the Killing form is a negative deﬁnite form on SU.2/. Owing to the factor of i deﬁning f2 and f3 , the Killing form is a form of signature .2; 1/ on SL.2; R/. e Killing form is a non-degenerate bi-invariant pseudo-Riemannian metric. Consequently, by eorem 6.19, the metric exponential map and the group exponential map coincide. erefore, all geodesics extend for inﬁnite time and SL.2; R/ is geodesically complete with respect to the metric deﬁned by the Killing form. By eorem 6.21, if sl.2; R/ is a Lie subalgebra of so.m/ for some m, then the Killing form is negative semi-deﬁnite. Since in fact the Killing form has signature .2; 1/, sl.2; R/ is not a Lie subalgebra of so.m/ for any m. Let be a representation of SL.2; R/ to SO.m/ for some m. Let fQi D fi . e fQi satisfy the bracket relations of the fi since is a Lie algebra homomorphism. Since is not injective, we have a non-trivial relation of the form x WD a1 fQ1 C a2 fQ2 C a3 fQ3 D 0 :

80

6. LIE GROUPS

Suppose a1 ¤ 0. We have ad.fQ3 / ad.fQ2 /x D 4a1 fQ1 . erefore, fQ1 D 0. Consequently, rangefad.fQ1 /g D 0

so fQ2 D 0

and fQ3 D 0 :

erefore, is the zero map. e argument is the same with the remaining coeﬃcients. Since is the zero map, is the trivial representation by Lemma 6.2. is proves Assertion 2. Let D a1 f1 C a2 f2 C a3 f3 2 g. en 2 D h; i Id. Set D h; i. Note that 3 D , 4 D 2 and so forth. is yields the following consequences. p 1. If < 0, let D . en: 1 1 2 1 1 2 exp./ D .1 C 2Š C 4Š C / Id C.1 C 3Š C 5Š C / 1 4 1 5 1 2 1 3 1 C / D .1 2Š C 4Š C / Id C . 3Š C 5Š 1 D cos./ Id C sin./ :

2. If D 0, then exp./ D Id C . p 3. If > 0, let D . en: 1 1 1 2 1 2 exp./ D .1 C 2Š C 4Š C / Id C.1 C 3Š C 5Š C / 1 2 1 4 1 3 1 5 1 D .1 C 2Š C 4Š C / Id C . C 3Š C 5Š C / 1 D cosh./ Id C sinh./ :

is shows exp./ 2 spanfId; g. Let g WD

Id Cf1 C f2 D

1 0

2 1

!

:

As f1 C f2 is a null vector, exp.t/ D Id Ct ¤ Id for any t so g … rangefexpg g. is proves Assertion 3. By Lemma 6.11, 1 .SL.2; R// D 1 .SO.2// D 1 .S 1 / D Z. is also follows as we can identify SL.2; R/ with the pseudo-sphere .x 0 /2 C .x 1 /2

.x 2 /2

.x 3 /2 D 1

and this deformation retracts onto the circle .x 0 /2 C .x 1 /2 D 1, x 2 D x 3 D 0. We now use the fact that sl.2; R/ ˝R C D su.2/ ˝R C which was observed above. Let be the covering projection from SL.2; R/ to SL.2; R/. We use to identify the Lie algebra of SL.2; R/ with sl.2; R/. Let be a representation of SL.2; R/ to GL.m; R/ for some m and let be the associated Lie algebra homomorphism from sl.2; R/ to gl.2; R/. We complexify to regard c as a Lie algebra homomorphism from sl ˝R C to gl.m; C/. We may then restrict to regard c as a Lie algebra homomorphism su.2/ from su.2/ to gl.m; C/. Since SU.2/ is compact and simply connected, su.2/ arises from a representation SU.2/ from SU.2/ to GL.2; C/.

6.9. THE COHOMOLOGY OF COMPACT LIE GROUPS

81

3

By the Peter–Weyl eorem, every ﬁnite-dimensional representation of S is unitarily equivalent to a subrepresentation of L2 . We apply eorem 5.12 to ﬁnd an orthogonal direct sum decomposition of L2 .S 3 / D ˚j 2N H.4; j / where H.4; j / is the subspace of homogeneous polynomials of degree j which are harmonic with respect to the Euclidean Laplacian on R4 . e decomposition of L2 is equivariant with respect to the isometry group of S 3 . Consequently, every ﬁnite-dimensional representation of S 3 is equivalent to a subrepresentation of ˚j j0 H.4; j /. In particular, it is polynomial in the standard coordinates on R4 , i.e., in the coordinates x i . erefore, it extends to a map SU.2/ from SU.2/ to GL.2; C/. Since c .x x/ Q D c .x/ c .x/ Q , the same identity holds for complex coordinates and, consequently, c is a representation from SUC .2/ to GL.2; C/. Restricting to SL.2; R/, we see that in fact arises from a representation of SL.2; R/. Consequently, kerfg kerfg and is not a faithful representation of SL.2; R/. is shows that SL.2; R/ is not a matrix group. t u

6.9

THE COHOMOLOGY OF COMPACT LIE GROUPS

Let g be the Lie algebra of a compact connected Lie group; g is vector space of left-invariant vector ﬁelds on G . Let Œ; be the Lie bracket. Let g be dual space; this is the vector space of left-invariant 1-forms. e exterior derivative and the Lie bracket are related by the formula: d!Œx; y D

Œx; y.!/ :

Let fei g be a basis for g and let fe i g be the corresponding dual basis for g . We expand Œei ; ej D cij k ek

to deﬁne the Lie algebra structure constants of Equation (6.2.b). We have: de k D

cij k e i ^ e j :

We consider the ﬁnite cochain complex fp .g /; d g and let H .g ; d / be the associated cohomology. e following result of Chevalley and Eilenberg [14] provides a method for computing the de Rham cohomology of G using Lie theoretic methods. e proof we give is based on the Hodge Decomposition eorem (see eorem 5.13); it is also possible to proceed directly. eorem 6.26

Let G be a compact connected Lie group which is equipped with a bi-invariant metric

of total volume 1. 1. If !p 2 kerfp g, then !p is bi-invariant. 2. If !p 2 p .g / and if !p D d p p 1 .g / so that !p D d p 1 .

1

for some

p 1

in C 1 .p

1

G/, then there exists p

1

in

3. e inclusion map i W .p .g /; d / ! .C 1 .p G/; d / is a chain map which deﬁnes an isomorp phism from H p .g ; d / to HdR .G/.

82

6. LIE GROUPS

Proof. Let ! be a p -form with p D 0, and let h 2 G . Because the metric is left-invariant, Lh D Lh . erefore, Lh ! 2 kerf p g as well. On the other hand, G is assumed to be connected. Consequently, there is a smooth curve .t / connecting h to the identity and providing a homotopy between Lh and LId . e homotopy axiom then shows ŒLh ! D Œ! in de Rham cohomology. e Hodge Decomposition eorem (see eorem 5.13) then shows Lh ! D ! . e argument is the same for right-invariance; Assertion 1 now follows. Suppose ! is left-invariant and that ! D d for some smooth .p 1/-form which is not necessarily left-invariant. We average over the group to see that: Z Z Z ! D .Lh !/ dvol.h/ D .Lh d / dvol.h/ D d .Lh / dvol.h/ G G G Z D d f g where WD .Lh / dvol.h/ : G

Assertion 2 now follows by noting is left-invariant. We have used, of course, the fact that we can interchange the order of diﬀerentiation and integration. Assertion 1 shows that Œi is surjective p .G/. Assertion 3 now and Assertion 2 shows that Œi is injective as a map from H p .g ; d / to HdR t u follows. 6.9.1 THE HOPF STRUCTURE THEOREM. e results in this section arise from work of H. Hopf [33].

H. Hopf (1894–1971) We refer the reader to the discussion in Section 8.1.6 for the deﬁnition of a unital graded ring. We say that a connected unital graded ring R is a co-ring if we have a co-multiplication which is a graded ring morphism from R to R R which is co-associative, i.e., .Id / D .

Id/ :

We do not assume the co-multiplication is co-commutative. We say that R is a Hopf algebra if, in addition, there is an augmentation " W R ! F so .Id / D Id

and

.

Id/ D Id :

We use the existence of a co-unit to pin down slightly. We expand .a/ D 1

bn C stuﬀ C an

1

where the deleted material “stuﬀ ” belongs to i >0;j >0;i Cj Dn Ri

Rj . We conclude

6.9. THE COHOMOLOGY OF COMPACT LIE GROUPS

83

.a/ D a ˝ 1 C stuﬀ C 1 ˝ a .

We have the following examples of Hopf algebras.

2 1. Let RŒx2 D R ˚ R x2 ˚ R x2 ˚ be the polynomial ring on an indeterminate x2 of even degree 2 > 0. Deﬁne .x/ D x ˝ 1 C 1 ˝ x ; this is then a Hopf algebra.

2. Let Œy2C1 D R ˚ R y2C1 be the exterior algebra on a generator y2C1 of odd degree 2 C 1 1. 3. e tensor product of Hopf algebras is again a Hopf algebra. erefore, R WD RŒx21 ; : : : ; x2k ˝ Œy21 C1 ; : : : ; y2` C1

is a ﬁnitely generated Hopf algebra. Let B be a Hopf algebra and let A be a Hopf subalgebra. Assume that A and some other element x … A of positive degree generate B as an algebra. en: ( ) Œx if deg.x/ is od d BD ˝R A . RŒx if deg.x/ is eve n P Proof. Let AC D ˚j >0 Aj be the elements of positive order. Let I WD x a for a 2 A and deg.a / > 0. is is an ideal of B and x … I . Let Lemma 6.27

W B ! B=I

and

D .Id ˝/ ı W B ! B ˝ B=I :

We may decompose .x/ D x ˝ 1 C stuﬀ C 1 ˝ .x/. Since the “stuﬀ ” lies in Bi ˝ Bj for i > 1, j > 1, and i C j D deg.x/, these elements must belong to AC and, consequently, these elements are all killed by . Consequently, .x/ D x ˝ 1 C 1 ˝ .x/. Because A is a Hopf subalgebra, .a/ D a ˝ 1 if a 2 AC . Let deg.x/ D n. Suppose we had a relation of the form a1 C a2 x D 0 for the ai 2 A and a2 ¤ 0. en ai 2 AC since x … A. We compute 0 D .a1 C a2 x/ D .a1 / C .a2 / .x/ D .a1 C a2 x/ ˝ 1 C a2 ˝ .x/ D a2 ˝ .x/ :

is implies a2 D 0 since .x/ ¤ 0. If n is odd, x 2 D 0 so B D A ˚ A x and we are done. erefore, we may suppose that n is even. Suppose we have a non-trivial relation a0 C C a x D 0 :

Choose minimal with 2. Apply to see .a / D a ˝ 1. If deg.a / D 0, then a is a scalar and this is automatic. If deg.a / > 0, this also is automatic. X a .x ˝ 1 C 1 ˝ .x// D 0 :

P We compare graded degrees to see a x istic zero. is gives a relation of lower degree.

1

˝ .x/ D 0. Fortunately we are in characteru t

84

6. LIE GROUPS

eorem 6.28 If R is a ﬁnitely generated Hopf algebra, then R is isomorphic as a graded associative ring to RŒx21 ; : : : ; x2k ˝ Œy21 C1 ; : : : ; y2` C1 . If R is ﬁnite-dimensional, then there is no

polynomial component.

Proof. We suppose B is a ﬁnitely generated Hopf algebra. We choose generators fx1 ; : : : ; xn g

so

deg.xi / deg.xiC1 / :

Let B be the subalgebra generated by fx1 ; : : : ; x g. en BC1 is generated by B and xC1 . Furthermore, .x / D x ˝ 1 C stuﬀ C 1 ˝ x so counting degrees the stuﬀ belongs to B . erefore, B is a Hopf algebra. We apply Lemma 6.27 and induction. t u We use eorem 6.28 to see: eorem 6.29 If G is a compact connected Lie group, then HdR .G/ is isomorphic to an exterior algebra Œx2i1 1 ; : : : ; x2i` 1 on a ﬁnite number of odd-dimensional generators. p Proof. Let G be a compact connected Lie group. By eorem 5.7, HdR .G/ is ﬁnite-dimensional p for 0 p m; HdR .G/ D 0 for p > m. Let m W G G ! G be the multiplication in the group. is induces a ring homomorphism m W HdR .G/ ! HdR .G G/. By the Künneth formula, we have a natural graded ring isomorphism 1 ^ 2 W HdR .G/ ˝ HdR .G/ ! HdR .G G/ : us, WD .1 ^ 2 / 1 ı m provides a co-multiplication from HdR .G/ to HdR .G G/. Let be the inclusion of the identity of the group into G . Dually, this gives a map from HdR .G/ 0 to HdR .G/ D R. A bit of diagram chasing shows that HdR is a Hopf algebra and the desired conclusion now follows from eorem 5.16. t u

6.10 THE COHOMOLOGY OF THE UNITARY GROUP We illustrate eorem 5.18 by taking G D U.n/ to be the complex unitary group. e Maurer– Cartan form [8] WD g 1 dg plays a crucial role. eorem 6.30 Set !2k 1 WD Trf.g 1 dg/2k 1 g. en !2k x2k 1 WD Œ!2k 1 is the corresponding cohomology class, then

Proof. We ﬁrst show d!2k d d!2k

1

1

is a closed .2k

HdR .U.n// ˝R C D C Œ!1 ; : : : ; !2n

D d.g

1

D 0. Let WD g

1

/ dg D

g

1

1

1/-form on U.n/. If

1 .

dg . en

dg g

1

dg D

2;

D d.Trf 2k 1 g/ D Trfd. 2k 1 /g D Trfd ^ 2k 2 ^ d ^ 2k 3 C g D Trf 2k g D Trf 2k 1 g D Trf 2k 1 g D Trf 2k g :

6.10. THE COHOMOLOGY OF THE UNITARY GROUP

85

It now follows that d!2k 1 D d!2k 1 and hence d!2k 1 D 0. We are using the fact, of course, that is a matrix of 1-forms. e 1-forms anti-commute and the trace commutes. Let in 1 be the inclusion of U.n 1/ in U.n/. We have ! ! 1 T 0 g dg 0 n 1 n 1 in 1 .T / D ; gn 1 dgn .in 1 .T // D ; 0 Id 0 0 ! n2k 1 1 0 2k 1 in 1 .n / D ; in 1 .Trfn2k 1 g/ D Trfn2k 1 1 g : 0 0 erefore, these classes are universal and we do not need to indicate the dimension n explicitly. By Lemma 5.10, there exists g W S 2n 1 ! U.2n / so that Z g !2n 1 ¤ 0 : S 2n

e ﬁbration U.`

1/ ! U.`/ ! S 2` 0 ! 2n

1 .U.`

1

1

yields a sequence:

1// ! 2n

1 .U.`//

for ` > n :

!0

Consequently, the map 2k 1 .U.k// ! 2k 1 .U.2k // is injective. erefore, in fact, after “smoothing” we can push g down to a map from S 2n 1 to U.n/. is yields maps f from S 2 1 to U.k/ for 1 k so that Z S 2 2 So none of these classes is zero in HdR ? HdR .U.k// D .y21

If i , then f y2i f fŒy21

1

1

1

D0

1 ; : : : ; y2` 1 /

so

Consequently, we may assume that Œ!2 0 ¤ y21

1

1

¤ 0:

. eorem 6.29 shows

D 0 in HdR .S 2

1 ; : : : ; y2i 1 g

f !2

1

1

where 1 1 ` :

/. Consequently,

Œ!2

1

… Œy21

1 ; : : : ; y2i 1

if

i :

is taken as one of the generators. Since

^ ^ y2`

1

in HdR .U.k// ;

21 1 C C 2` 1 dimfU.k/g. Since 1 C 3 C C 2k other generators than the Œ!2 1 . Consequently, as desired, HdR .U.k// D .Œ!1 ; : : : ; Œ!2k

1 / .

1 D dimfU.k/g, there are no t u

87

CHAPTER

7

Homogeneous Spaces and Symmetric Spaces Good auxiliary references for the material of Chapter 7 are Arvanitoyeorgos [2], Helgason [30], O’Neill [49], Warner [60], and Ziller [64]. In Section 7.1, we prove results about smooth structures on coset spaces. In Section 7.2, we discuss the isometry group of a pseudo-Riemannian manifold. In Section 7.3, we treat Killing vector ﬁelds. In Section 7.4, we establish facts concerning homogeneous pseudo-Riemannian manifolds. In Section 7.5, we use Jacobi vector ﬁelds to study the geometry of local symmetric spaces. In Section 7.6, we examine symmetric spaces.

7.1

SMOOTH STRUCTURES ON COSET SPACES

Let H be a Lie group which acts continuously on a topological space X from the right. If x 2 X , let Œx WD x H and let W x ! Œx be the natural projection from X to the set of right cosets X=H . Give X=H the quotient topology; a subset U of X=H is open if and only if 1 .U / is open in X . It is immediate that is continuous; the group action can be used to show that is an open map. If f W X ! Y satisﬁes f .x h/ D f .x/ for all x 2 X and all h 2 H , let Œf W X=H ! Y be the induced map on the coset space; Œf W X=H ! Y is continuous if and only if f W X ! Y is continuous. We say that F ! E !M is a ﬁber bundle if is a smooth map from a manifold E onto a manifold M , and if there is a cover of M by open sets O˛ such that there exist ﬁber preserving diﬀeomorphisms from 1 .O˛ / to O˛ F . We may identify F with 1 .P / for any P 2 M . e ﬁber bundle is said to be a principal bundle if F is a Lie group which acts smoothly on E without ﬁxed points from the right and if M D E=F is the associated coset space; see the discussion in Section 2.1.9 of Book I for further details. eorem 7.1

Let H be a closed subgroup of a Lie group G .

1. G=H has a natural smooth structure so that H ! G !G=H is a principal H -bundle and so that the left action of G on G=H is smooth. 2. If H is a normal subgroup of G , then the coset space G=H is a Lie group. 3. If is a group homomorphism from G onto a Lie group GQ so that .H / is a Lie subgroup of GQ , Q then induces a smooth map from G=H onto G=.H /.

88

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Proof. If Œx D Œy in G=H , then ŒLg x D ŒLg y. Let ŒLg Œx WD ŒLg x deﬁne a continuous left action of G on G=H . Let h and g be the Lie algebras of H and of G , respectively. Use a rightinvariant Riemannian metric on G to decompose g D h ˚ h? . Let B;h (resp. B;h? ) be the open ball of radius > 0 about the origin in h (resp. h? ). Let .h ; h? / WD exp.h? / exp.h /

for

h 2 B;h

and h? 2 B;h? :

en is a diﬀeomorphism from B;h B;h? to a neighborhood of the identity in G if is suﬃciently small. Let O be an open subset of B;h? . e following are open sets of G : exp.O/ exp.B;h / D .B;h O/; 1 f.exp.O//g D exp.O/ H :

exp.O/ H D exp.O/ exp.B;h / H;

erefore, 1 f.exp.O//g is an open subset of G so ı exp is a continuous open map from B;h? onto the open neighborhood BQ WD . ı exp/.B;h? / of ŒId. If . ı exp/.1;h? / D . ı exp/.2;h? /

for

i;h? 2 B;h? ;

then exp.1;h? / D exp.2;h? / h for some h 2 H . If is suﬃciently small, then h D exp.h / for h close to 0 and exp.1;h? / D .2;h? / exp.h / for some h 2 h close to 0. is implies .0; 1;h? / D .h ; 2;h? /

and, therefore, 1;h? D 2;h? . Consequently, ı exp is a homeomorphism from B;h? to BQ which deﬁnes a coordinate chart near ŒId. If Œx 2 BQ , let h? W Œx ! . ı exp/

1

(7.1.a)

.Œx/ 2 B;h?

be the inverse homeomorphism. We deﬁne a continuous section s to over BQ by setting: s.Œx/ WD exp.h? .Œx// W BQ ! exp.B;h / G :

(7.1.b)

We then have that .s.Œx// D Œx for Œx 2 BQ . We use ŒLg to translate the chart BQ and thereby obtain charts covering G=H as g 2 G . If h? is orthogonal projection on h? , then the transition function relating the given chart to the translated chart is given by the smooth function h? ! h? .

1

.Lg .exp.h? //// :

erefore, G=H admits a smooth manifold structure, ŒLg is a smooth left action of G on G=H , and s is a smooth section to over BQ . If y 2 1 .BQ / D B;h? H , then we may express y D s.Œy/ h.y/

where h.y/ WD s.Œy/

1

y:

e map y ! .Œy; h.y// is an H -equivariant diﬀeomorphism from 1 .BQ / onto BQ H ; the inverse map is .Œy; h/ ! s.Œy/ h. Consequently, H ! G !G=H is a principal H -bundle. is proves Assertion 1.

7.1. SMOOTH STRUCTURES ON COSET SPACES

89

If H is a normal subgroup of G , then G=H is a group. Let si be local smooth sections to which are deﬁned near Œxi . As Œx1 Œx2 D Œs.Œx1 / s.Œx2 /

and

Œx1

1

D Œs.Œx1 /

1

;

the group multiplication and inversion are smooth. erefore, G=H is a Lie group; this proves Assertion 2. Assertion 3 is immediate from the discussion given above. t u Let G be a Lie group which acts smoothly and transitively on a manifold M from the left. Fix P 2 M . Let H D HP WD fg 2 G W g P D P g be the isotropy subgroup; H is a closed subgroup of G . Let W g ! g P ; is a smooth map from G onto M . If h 2 H , then .g h/ D g h P D g P D .g/

so induces a continuous map Œ W G=H onto M . We show that Œ is 1-1 as follows: .g1 / D .g2 / , g1 P D g2 P , g2 1 g1 2 H

, g2 1 g1 P D P , g1 2 g2 H

, Œg1 D Œg2

in G=H :

eorem 7.2 Let G be a Lie group which acts smoothly and transitively on a manifold M from the left. en Œ is a G -equivariant diﬀeomorphism from G=H to M .

Proof. Since Œ is G -equivariant, it suﬃces to show that Œ is a diﬀeomorphism near ŒId. We adopt the notation established in the proof of eorem 7.1. Let h? W BQ ! B;h? be the diﬀeomorphism of Equation (7.1.a) and let s.Œx/ D exp.h? .Œx// be the section of Equation (7.1.b). Since ŒŒx D s.x/ P , Œ is smooth. For 0 ¤ h? 2 B;h? , let .t; QI h? / WD exp.th? / Q

deﬁne a 1-parameter ﬂow on M with associated vector ﬁeld Xh? . We have Xh? .P / D @ t fexp.th? / P gj tD0 D .Œ ı ı exp /fh? g :

If Xh? .P / D 0, then the Fundamental eorem of ODEs shows .t; P I h? / D P for all t . However, .t; P I h? / D .Œ ı ı exp/.t h? / ¤ P if t ¤ 0 since Œ is 1-1 from BQ to M and since ı exp is a diﬀeomorphism from B;h? to BQ . Consequently, fŒ ı ı expg is an injective map from T0 B;h? to TP M and hence Œ is a diﬀeomorphism from a neighborhood of ŒId in G=H to a neighborhood of P in M . Using left translation in the group, we see the same is true for any element of G=H and, therefore, Œ is a global diﬀeomorphism. t u

90

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Let A and B be closed subgroups of a Lie group C with A B C . en the following sequence is a smooth ﬁber bundle: eorem 7.3

C =A!C =B

B=A ! C =A

!

C =B .

Proof. Let x 2 C . By eorem 7.1, there exists an open neighborhood O of the coset ŒxC =B in C !C =B C =B and a smooth section s to the ﬁbration B ! C ! C =B deﬁned on O. e maps W .ŒxC =B ; b/ ! s.ŒxC =B / b

and

1

W x ! .ŒxC =B ; s.ŒxC =B /

1

x/

1 are diﬀeomorphisms between O B and C !C .O/ in C . e maps =B 1 W .ŒxC =B ; ŒbB=A / ! s.ŒxC =B / ŒbB=A C =A ; and 1 1 W ŒxC =A ! C =A!C =B .ŒxC =A /; s.C =A!C =B .ŒxC =A // 1 C =A!C =B .ŒxC =A / 1 are diﬀeomorphisms between O ŒB=A and C =A!C .O/ in C =A that provide the required =B local trivialization of the natural map from C =B to C =A. t u

We have the following examples. e special orthogonal group SO.m C 1/ acts transitively on the unit sphere S m in RmC1 by isometries with isotropy subgroup SO.m/; SO.m/ ! SO.m C 1/ ! S m is a principle SO.m/ bundle. e unitary group U.m/ acts transitively on the unit sphere S 2m 1 in C m by isometries with isotropy subgroup U.m 1/; U.m 1/ ! U.m/ ! S 2m 1 is a principle U.m 1/ bundle. e general linear group GL.m C 1; R/ acts naturally and transitively on RmC1 f0g with the isotropy subgroup GL.m; R/; GL.m; R/ ! GL.m C 1; R/ ! RmC1 f0g is a principle GL.m; R/ bundle. Complex projective space CP m 1 D S 2m 1 =S 1 D U.m/= U.m 1/ U.1/. We take A D U.m

1/;

B D U.m

1/ U.1/;

C D U.m/

in eorem 7.3 to obtain the Hopf ﬁbration: S 1 D U.m 1/ U.1/= U.m 1/ D B=A ! S 2m 1 D U.m/= U.m 1/ D C =A ! CP m 1 D U.m/= U.m 1/ U.1/ D C =B . e Grassmannian Grk .V / is the set of all k -dimensional subspaces of a real or complex vector space V . Let .; / be a positive deﬁnite inner product on V . If 2 V , let . / denote orthogonal projection on V . is identiﬁes Grk .V / with the set of orthogonal projections of rank k in Hom.V /. e orthogonal group O.V; .; // acts transitively on Grk .V / and the isotropy subgroup is O./ O. ? /. We have Grk .Rm / D O.m/=fO.k/ O.m k/g; Grk .C m / D U.m/=fU.k/ U.m k/g :

7.2. THE ISOMETRY GROUP mC1

91

mC1

We take k D 1 and V D R (resp. V D C ) to obtain real (resp. complex) projective space m m RP (resp. CP ). e earliest work on these spaces is due to Julius Plücker and the more general framework is due to Hermann Grassmann.

H. Grassmann (1809–1877)

7.2

J. Plücker (1801–1868)

THE ISOMETRY GROUP

A diﬀeomorphism of a smooth pseudo-Riemannian manifold .M; g/ such that .g/ D g is said to be an isometry. Such maps form a group under composition and we let I .M; g/ be the group of isometries. e following is a useful remark. Let 1 and 2 be isometries of a connected pseudo-Riemannian manifold .M; g/. If there exists P 2 M so 1 .P / D 2 .P / and . 1 / .P / D . 2 / .P /, then 1 D 2 .

Lemma 7.4

Proof. Set WD 1 1 2 . en .P / D P and .P / D Id. Let

S D fQ 2 M W .Q/ D Q and .Q/ D Idg :

en S is a non-empty closed subset of M . Since .Q/ D Id, preserves the geodesics through Q and hence D Id near Q. Consequently, S is open. Since M is connected, S D M so D Id. u t g

Lemma 7.5 Let P be a point of a connected Riemannian manifold .M; g/. Use expP to identify g B5r .0/ in TP M with a neighborhood of P in M for some r > 0.

1. Let gs for s 2 Œ 1; 1 be a continuous 1-parameter family of Riemannian metrics on M gs g with g0 D g . en dM .P; / converges uniformly to dM .P; / on Brg .P / as s ! 0. 2. lim kt t!0

g tk fdM .t; t/g

1

D 1 for all ; 2 TP M .

3. If is a map from M to M which preserves distance, then 2 I .M; g/. Proof. Let 0 < < 51 be given. Since BN 3r .P / is a compact subset of B5r .P /, we may choose ı > 0 so jtj < ı and jsj < ı implies .1

/gs < g t < .1 C /gs

on B3r .P / :

(7.2.a)

92

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Let Q 2 Br .P /. Let ˛0 be the g0 geodesic from P to Q. en gt dM .P; Q/ Lgt .˛0 / .1 C /Lg0 .˛0 / 65 r :

Let ˛ t be a curve in M from P to Q with jLgt .˛ t / Lg t .˛ t /

(7.2.b)

gt dM .P; Q/j < r . We then have that

gt r dM .P; Q/ :

(7.2.c)

g0 g0 Suppose ˛ t leaves B2r .P /. Let ˇ t be the part of ˛ t from Q to the boundary of B2r .P /. en 6 r 5

gt dM .P; Q/ Lg t .˛ t / r Lg t .ˇ t / r .1 .1 /2r r D .2 3/r .2 53 /r D 75 r

/Lg0 .ˇ t /

r

g which is false. erefore, ˛ t remains in B3r .P /. Equations (7.2.a), (7.2.b), and (7.2.c) yield: gt dM .P; Q/ Lg t .˛ t / r .1 /Lgs .˛ t / gs r : dM .P; Q/ 11 5

r .1

gs /dM .P; Q/

r

Interchanging the roles of t and s permits us to establish Assertion 1 by estimating: gt .P; Q/ jdM

gs .P; Q/j dM

11 r 5

.

We work in normal coordinates. Let h t;ij .Q/ WD gij .tQ/. en h0 is the constant linear inner product given by g.0/ on TP M . Let t .Q/ D tQ be a dilation; t g D t 2 h t for t > 0. We derive Assertion 2 from Assertion 1 by computing: lim kt

t !0

g tk fdM .t; t /g

1

lim tdhM0 .; / ft dhMt .; /g

1

lim dhM0 .; / fdhMt .; /g

D 1:

D

t!0

D

t!0

1

We now prove Assertion 3. Let be a distance preserving map from M to M ; is continuous since d.P; Q/ < implies d. .P /; .Q// < . Fix P 2 M . Let PQ WD .P / and let Q < ), then there is a unique shortQQ WD .Q/. Choose > 0 so if d.P; Q/ < (resp. d.PQ ; Q/ Q Q est geodesic (resp. Q ) from P to Q (resp. P to Q). Let ı D d.P; Q/ < . Let R D .t / for 0 < t < ı and let RQ WD .R/. en d.P; R/ D t

and d.Q; R/ D ı

t:

Q D t , d.R; Q Q/ Q D ı t , and d.PQ ; Q/ Q D ı . e properties of the miniApplying yields d.PQ ; R/ mizing geodesic for Riemannian geometry which were established in Book I now imply RQ D .t Q /. is shows that maps geodesics to geodesics. We use the exponential map to identify a neighborhood of 0 in TP M with a neighborhood of P in M and similarly to identify a neighborhood of 0 in TPQ M with a neighborhood of PQ in M and to express D expPQ ı ı logP where is continuous map from .TP M; 0/ to .TPQ M; 0/;

7.2. THE ISOMETRY GROUP

93

the above analysis shows that .tv/ D t .v/ for any t 2 R and that k .v/k D kvk. We use the polarization identity, Assertion 1, and the fact that is distance preserving to compute: 2.;/ .P / kk kk

D D

kk2 Ckk2 k k2 .P / kk kk

Q 2 Ckk kk Q 2 Q .P / Q kk kk Q

D

kk2 Ckk2 .P / kk kk

Q Q 2 lim d.tQ ;t / .PQ / kt k Q t!0 kt k

D

2

d.t ;t / lim ktk .P / ktk

t!0

Q 2 Ckk kk Q 2 kQ k Q 2 Q .P / Q kk kk Q

D

Q / 2.; Q .PQ / : Q kk kk Q

Q / is shows that .; / D .; Q as well. But C is determined by fkk; kk; .; /g. is shows . C / D ./ C ./. Consequently, is an orthogonal transformation and is a smooth local isometry. Since preserves distances, is injective. Since is a local isometry, rangef g is open. Since is distance preserving, rangef g is closed. Since rangef g is non-empty and M is connected, rangef g D M . It now follows is an isometry. t u eorem 7.6

1. If .M; g/ is a pseudo-Riemannian manifold, then I .M; g/ is a Lie group and the natural action of I .M; g/ on M is smooth. A group homomorphism f from R to I .M; g/ is smooth if and only if the map .t; x/ ! f .t/x from R M to M is smooth. 2. Every compact Lie group can be realized as the full group of isometries of a compact Riemannian manifold. 3. If .M; g/ is a compact Riemannian manifold, then I .M; g/ is compact. Proof. We refer to Myers and Steenrod [45] and to Palais [50] for the proof of Assertion 1, and to Saerens and Zame [55] for the proof of Assertion 2 as these results are beyond the scope of this book. We establish Assertion 3 as follows. If .M; g/ is a compact Riemannian manifold, then I .M; g/ has the metric topology given by g g dM . 1 ; 2 / WD max dM . 1 P; 2 P / : P 2M

Find a countable dense subset fS1 ; S2 ; : : : g of M . Let n be a sequence of isometries. By passing to a subsequence, we may assume limn!1 n Si exists for every i . Let > 0 be given. As M is compact, there exists n D n./ so M D B .S1 / [ [ B .Sn / :

Choose N D N./ so that if a; b > N , then d. a Si ; b Si / < for 1 i n. Given P 2 M , choose i with 1 i n so d.P; Si / < . If a; b > N , we may estimate: d. a P; b P / d. a P; a Si / C d. a Si ; b Si / C d. b Si ; b P / d.P; Si / C C d.Si ; P / 3 :

is shows d. a ; b / < 3 if a; b > N . Consequently, the sequence f n g is uniformly Cauchy and converges uniformly to a continuous distance preserving map from M to M . Lemma 7.5 implies 2 I .M; g/. is shows I .M; g/ is compact. t u

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7.2.1 EXAMPLE. Assertion 3 of eorem 7.6 fails in the higher signature setting. Let .x 1 ; x 2 ; y 1 ; y 2 / be the usual periodic parameters on the 4-dimensional torus T 4 WD R4 =Z4 . Let ds 2 D dx 1 ı dy 1 C dx 2 ı dy 2 be a ﬂat metric of signature .2; 2/. e torus is a compact Abelian Lie group under addition and ds 2 is a bi-invariant metric on T 4 . ere is a canonical action of SL.2; Z/ on T 4 . If A 2 SL.2; Z/, we set .A/ D A ˚ .A 1 /t to deﬁne a map from SL.2; Z/ to SL.4; Z/ and thereby deﬁne an action of SL.2; Z/ on T 4 . If A D .nij /, then A .x 1 ; x 2 ; y 1 ; y 2 / WD .n11 x 1 C n12 x 2 ; n21 x 1 C n22 x 2 ; n22 y 1

n21 y 2 ; n12 y 1 C n11 y 2 / :

It is clear that A acts by isometries on T 4 . e fundamental group of T 4 is Z4 and the induced action of A on the fundamental group is given by . Consequently, all the isometries .A/ belong to diﬀerent arc components of I .T 4 ; ds 2 / so I .T 4 ; ds 2 / has an inﬁnite number of arc components. We refer to work of Melnick [42] to see that there exist compact Lorentzian manifolds such that the connected component of the isometry group is non-compact. 7.2.2 EXAMPLE. Give the upper-half plane H2 WD f.x; y/ 2 R2 W y > 0g the hyperbolic metric ds 2 WD y 2 .dz ı d z/ N . Let

PSL.2; R/ WD SL.2; R/=Z2 WD f˙1g be the projective special linear group. e linear fractional transformations az C b TA .z/ WD cz C d

for

AD

a b c d

!

2 SL.2; R/

deﬁne an action of PSL.2; R/ on H by orientation preserving isometries and every orientation preserving isometry of H2 arises in this way. If we identify H2 with the unit pseudo-sphere in Minkowski space, then we identify PSL.2; R/ with the connected component of the identity in O.1; 2/. PSL.2; R/ acts transitively on the unit tangent bundle of the hyperbolic plane by isometries and we may identify PSL.2; R/ with the unit sphere bundle of the tangent bundle to H2 . We showed in Lemma 6.25 that the Killing form is a bi-invariant Lorentzian metric on PSL.2; R/. Let ˙ be an orientable Riemann surface with a hyperbolic metric of constant Gaussian curvature 1. e universal cover of ˙ is then H2 and the deck group is a cocompact lattice in PSL.2; R/. We then have that M WD PSL.2; R/= is a compact manifold which admits a Lorentzian metric which is invariant under the left action of PSL.2; R/. erefore, PSL.2; R/ acts by isometries on .M; g/ and the natural map from PSL.2; R/ to I .M; g/ is smooth and nontrivial. If I .M; g/ is a compact Lie group, then by the Peter–Weyl eorem, the matrix coeﬃcients of the unitary representations of I .M; g/ separate points of I .M; g/. is would imply that there exists a non-trivial orthogonal representation of PSL.2; R/ which contradicts Lemma 6.25.

7.3. THE LIE DERIVATIVE AND KILLING VECTOR FIELDS

7.3

95

THE LIE DERIVATIVE AND KILLING VECTOR FIELDS

We present concepts originally introduced by the German mathematician Wilhelm Karl Joseph Killing and by the Norwegian mathematician Sophus Lie.

W. Killing (1847–1923)

S. Lie (1842–1899)

7.3.1 THE LIE DERIVATIVE. Let ˚ tX be the ﬂow for a vector ﬁeld X on M ; it is only locally deﬁned but this plays no role. Let S D Y ˝ 2 C 1 ..˝k TM / ˝ .˝` T M // be a mixed tensor ﬁeld where Y is a smooth section to ˝k TM and is a smooth section to ˝` T M . e Lie derivative LX of Y ˝ is deﬁned by setting:

LX .Y ˝ / WD LX .Y / ˝ C Y ˝ LX ./ where LX .Y / WD lim t 1 f.˚ Xt / Y Y g and LX ./ WD lim t t!0

t !0

It is then immediate that LX satisﬁes the Leibnitz formula:

1

f.˚ tX /

g :

LX .S1 ˝ S2 / D LX .S1 / ˝ S2 C S1 ˝ LX .S2 /

(7.3.a)

for any tensor ﬁelds S1 and S2 . If ˛p 2 C 1 .p T M / and if ˇq 2 C 1 .q T M /, then:

LX .˛p ^ ˇq / D LX .˛p / ^ ˇq C ˛p ^ LX .ˇq / and d LX .˛p / D LX .d˛p / :

If f 2 C 1 .M /, then LX .f / D X.f /. We now establish some results for the Lie derivative; there are many other identities but they can all be derived easily using the same techniques as we shall use to prove Lemma 7.7. Let X 2 C 1 .TM /, Y 2 C 1 .TM /, and ! 2 C 1 .T M /. en 1. X hY; !i D hLX Y; !i C hY; LX !i. 2. LX Y D ŒX; Y . 3.ŒLX ; LY D LŒX;Y .

Lemma 7.7

Proof. Suppose ﬁrst that X.P / ¤ 0. Choose local coordinates xE D .x 1 ; : : : ; x m / near P so X corresponds to @x 1 ; the ﬂow then takes the form ˚ tX .x/ E D .x 1 C t; x 2 ; : : : ; x m / :

Let Y D b j @x j and let ! D ck dx k . en:

f.˚ Xt / Y g.x/ E D b j .x 1 C t; x 2 ; : : : ; x m /@x j ; LX .Y / D [email protected] 1 b j [email protected] j ;

f˚ t !g.x/ E D ck .x 1 C t; x 2 ; : : : ; x m /dx k ;

LX .!/ D [email protected] 1 ck gdx k :

(7.3.b)

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7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Assertion 1 and Assertion 2 now follow at P . More generally, suppose X D ai @x i where a.P / is zero. We use Lemma 6.8 to see ˚ tX .x/ E D xE C t a.x/ E C 12 t 2 da.x/ E a.x/ E C O.t 3 / :

is shows that LX .Y /.P / and LX .!/.P / depend smoothly on the 1-jets of Y at P , the 1-jets of ! at P , and the 1-jets of X at P . If X vanishes identically near P , then ˚ tX .Q/ D Q for Q near P and LX .Y / and LX .!/ vanish identically near P ; Assertion 1 and Assertion 2 then hold trivially. If there exists a sequence of points Pn ! P with X.Pn / ¤ 0, then Assertion 1 and Assertion 2 hold at Pn and hence by continuity at P . is establishes Assertion 1 and Assertion 2 in generality. If Z is a smooth vector ﬁeld, then Assertion 2 and the Jacobi identity imply that: ŒLX ; LY Z D ŒX; ŒY; Z

ŒY; ŒX; Z D ŒŒX; Y ; Z D LŒX;Y .Z/ :

is proves Assertion 3 for vector ﬁelds; Assertion 3 for 1-forms then follows by duality from Assertion 1 and for general mixed tensor ﬁelds from Equation (7.3.a). t u 7.3.2 KILLING VECTOR FIELDS. Lemma 7.8 Let .M; g/ be a connected pseudo-Riemannian manifold. If X 2 C 1 .TM /, let AX .Y / WD rY X .

1. e following conditions are equivalent and if any is satisﬁed, then X is said to be a Killing vector ﬁeld: (a) LX .g/ D 0. (b) e local ﬂows ˚ tX deﬁned by X preserve the metric g . us, these local ﬂows are isometries. (c) AX is skew-symmetric. 2. If X is a Killing vector ﬁeld, then: (a) g.X..s//; .s// P is constant for any geodesic . (b) If there exists P 2 M so X.P / D 0 and so rX.P / D 0, then X 0. 3. If X and Y are Killing vector ﬁelds, then ŒX; Y is a Killing vector ﬁeld. Proof. If the local ﬂows ˚ tX preserve g , then LX .g/ D 0. Conversely, suppose that LX .g/ D 0. If X.P / ¤ 0, then we may choose local coordinates xE D .x 1 ; : : : ; x m / so X D @x 1 . e condition that LX .g/ D 0 then implies gij .x 1 ; : : : ; x m / D gij .x 2 ; : : : ; x m / so the ﬂows given in Equation (7.3.b) are isometries. e same continuity argument used to prove Lemma 7.7 can then be used to deduce this result even if X.P / D 0. is proves the equivalence of Assertion 1-a and Assertion 1-b. We prove the equivalence of Assertion 1-a and Assertion 1-c by using Lemma 7.7 to compute:

7.3. THE LIE DERIVATIVE AND KILLING VECTOR FIELDS

.LX g/.Y; Z/ D .LX g/.Y ˝ Z/ D X g.Y ˝ Z/ D X g.Y; Z/

g.ŒX; Y ; Z/

D .rX g/.Y; Z/

g.AX Y; Z/

D X g.Y; Z/

D

97

g.LX .Y ˝ Z//

g.ŒX; Z; Y /

g.rX Y; Z/ C g.rY X; Z/

g.AX Z; Y /

g.rX Z; Y / C g.rZ Y; X /

fg.AX Y; Z/ C g.AX Z; Y /g .

Let X be a Killing vector ﬁeld and let be a geodesic. By Assertion 1-c, AX is skewsymmetric so g.; P AX / P D 0. We establish Assertion 2-a by computing: @s g.; P X/ D g.rP ; P X / C g.; P rP X/ D 0

g.; P AX / P D 0.

Assertion 2-b is the analog of Lemma 7.4 on the inﬁnitesimal level. If X.P / D 0, then the map .˚ tX / from R to GL.TP M / is a group homomorphism and the corresponding representation on the Lie algebra level is given by @ tD0 .˚ tX / mapping the Lie algebra r of R to the Lie algebra gl of GL. Choose local coordinates xE D .x 1 ; : : : ; x m /. By Lemma 6.8, ˚ tX .x/ E D xE C ta.x/ E C O.t 2 /. erefore, .˚ tX / .P /.v/ D lim !0

1

f˚ tX .P C v/

D v C t lim !0

1

˚ tX .P /g

fa.P C v/

a.P /g C O.t 2 / .

Since rX.P / D 0, da.P / D 0 so .˚ tX / .P /.v/ D Id CO.t 2 /. is implies that @ t D0 f˚ tX g .P / D 0 :

Consequently, @ t D0 .˚ tX / is the trivial map and f˚ tX g.P / D Id for all t . As the maps ˚ tX are local isometries, they commute with expP , i.e., ˚ tX ı expP D expP ı.˚ tX / D expP :

erefore, ˚ t D Id near P so X D 0 near P . is proves Assertion 2-b. By Lemma 7.7, we have that ŒLX ; LY D LŒX;Y . Since LX .g/ D 0 and LY .g/ D 0, LŒX;Y g D 0. Assertion 3 follows. u t 7.3.3 THE LIE ALGEBRA OF A PSEUDO-RIEMANNIAN MANIFOLD. Denote the set of Killing vector ﬁelds by K D K.M; g/. By Lemma 7.8, K is a real vector space which is closed under bracket and forms a Lie algebra. Let t be a smooth 1-parameter family of isometries in I .M; g/; t is the 1-parameter family generated by a Killing vector ﬁeld. erefore, the Lie algebra of the group of isometries of I .M; g/ is a Lie subalgebra of K. However, every element of K need not correspond to a 1-parameter family of isometries. For example, if .M; g/ D ..0; 1/; dx 2 /, then X D @x is a Killing vector ﬁeld but I .M; g/ D Z2 is 0-dimensional. Lemma 7.9 Let .M; g/ be a pseudo-Riemannian manifold of dimension m. If M is compact or if .M; g/ is geodesically complete, then every Killing vector ﬁeld deﬁnes a 1-parameter subgroup of isometries of M and K is the Lie algebra of the isometry group I .M; g/.

98

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Proof. Let X 2 K. If M is compact, then the ﬂow ˚ t for X on M is globally deﬁned for all t and deﬁnes a 1-parameter ﬂow by isometries; this proves the Lemma in this case. Next, suppose .M; g/ is complete. We consider the forward ﬂow t 0 as the case t 0 can be handled by replacing X by X . For each point P 2 M , let .P / > 0 be chosen maximal so that ˚ t is deﬁned for 0 t < .P /. e maximal domain O for ˚ is

O WD [P 2M fŒ0; .P // P g : As lim infQ!P .Q/ D .P / ; O is open; ˚ is smooth on O and ˚s ˚ t D ˚sCt where deﬁned. We must show .P / D 1 for all P . Suppose to the contrary that D .P / < 1 for some P and argue for a contradiction. e exponential map expP is a diﬀeomorphism from a neighborhood of 0 in TP M to a neighborhood of P in M . Let logP be the local inverse. Choose 0 < ı 21 so that ˚ t .P / 2 domainflogP g for 0 t ı . Let ı t < . Since the maps ˚ t are local isometries, ˚ t ı expP D exp˚ t P ı.˚ t .P // . As .M; r/ is complete, we may express: ˚ t .P / D ˚ ı ˚ tCı P D ˚ ı expP flogP ˚ tCı D exp˚ ı P f.˚ ı .P // logP ˚ tCı P g :

e right-hand side of the equation is well-deﬁned and smooth for that Œ0; / was not a maximal domain for ˚ t .P /.

P g

ı t . is shows t u

Let I0 .M; g/ be the connected component of the identity in the isometry group I .M; g/. We say that a connected pseudo-Riemannian manifold .M; g/ is homogeneous if I0 .M; g/ acts transitively on M . We say that .M; g/ is locally homogeneous if given any pair of points P and Q in M , there exists an isometry from some neighborhood of P to some neighborhood of Q in M . Observe that if G is a connected Lie group which is equipped with a left-invariant non-degenerate bilinear form, then the left action of G is a transitive action by isometries and, consequently, .G; g/ is homogeneous. Lemma 7.10

Let .M; g/ be a pseudo-Riemannian manifold of dimension m.

1. If .M; g/ is homogeneous, then there are m linearly independent Killing vector ﬁelds at every point of M . 2. Assume that M is compact or that .M; g/ is complete. If there are m linearly independent Killing vector ﬁelds at every point of M , then .M; g/ is homogeneous. Proof. Assume .M; g/ is homogeneous so I0 .M; g/ acts transitively on M . By eorem 7.3, the map I0 .M; g/ ! I0 .M; g/ P is a submersion so the natural map from the Lie algebra of I0 .M; g/ to TP M is surjective for any P 2 M . Consequently, there exist m linearly independent Killing vector ﬁelds at P . Conversely, assume that there exist m linearly independent Killing vector ﬁelds at every point P of M . We apply Lemma 7.9 to see every Killing vector ﬁeld deﬁnes

7.3. THE LIE DERIVATIVE AND KILLING VECTOR FIELDS

99

a smooth 1-parameter family of isometries. It now follows that I0 .M; g/ P contains an open neighborhood of P . Since M is connected, it follows I0 .M; g/ acts transitively on M and .M; g/ is homogeneous. t u e hypothesis that M is either compact or complete is necessary to ensure that the conclusions of either Lemma 7.9 or Lemma 7.10 hold. e vector ﬁeld @x on ..0; 1/; dx 2 / is a globally deﬁned Killing vector ﬁeld but there is no corresponding 1-parameter family of global isometries of .0; 1/ and .0; 1/ is not globally homogeneous. Results of Kobayashi [37] and of Nomizu [48] provide a useful characterization of homogeneity (see also Hall [27] in the Lorentzian setting). We ﬁrst introduce a bit of notation. Let W D TM ˚ .T M ˝ TM /; this is a vector bundle of dimension m C m2 . Let X be a Killing vector ﬁeld deﬁned on an open subset O of M ; X will be called a local Killing vector ﬁeld. If P 2 O, set SX .P / WD X.P / ˚ rX.P / 2 WP .

Let KP D [X SX .P /. Since the sum of two Killing vector ﬁelds or a constant times a Killing vector ﬁeld is again a Killing vector ﬁeld, KP is a linear subspace of W . Let K WD [P KP . If .M; g/ is homogeneous, then dimfK./g is constant. e following result provides a partial converse; it is necessary to assume M simply connected to avoid problems with the fundamental group. eorem 7.11 Let .M; g/ be a simply connected pseudo-Riemannian manifold. Assume dimfK./g is constant on M . en K is a trivial vector bundle over M and every local Killing vector ﬁeld on M extends to a global Killing vector ﬁeld on M .

Proof. Let D dimfK./g. Let P be a point of M . Choose local Killing vector ﬁelds fX1 ; : : : ; X g so that fSX1 .P /; : : : ; SX .P /g forms a basis for KP . By continuity, fSX1 .Q/; : : : ; SX .Q/g are linearly independent elements of WQ for Q in some neighborhood O of P . Since dimfKQ g D , these elements span KQ and, therefore, fSX1 ; : : : ; SX g is a frame for K.O/. is shows that K.M / is a smooth vector subbundle of W . Let fY1 ; : : : ; Y g be another collection of local Killing vector ﬁelds such that fSY1 .P /; : : : ; SY .P /g

is a basis for KP . Express SYi .P / D aij SXj .P /. By Lemma 7.8, a Killing vector ﬁeld is determined by its value at a point and the value of its covariant derivative at that point. us, Yi D aij Xj and, consequently, SYi D aij SXj near P . is means that K is locally ﬂat, i.e., the transition functions of the vector bundle K with respect to this canonically deﬁned system of local frame ﬁelds are locally constant. Any locally ﬂat vector bundle over a simply connected space is trivial and the trivialization reﬂects the locally ﬂat structure. In other words, we may use the locally ﬂat structure to extend the Killing vector ﬁelds along curves; as M is simply connected, there is no holonomy and this gives a global extension. t u

100

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e hypotheses of eorem 7.11 are clearly necessary. In general, dimfKP g can vary with the point P . For example, let the torus T 2 be given the ﬂat Riemannian metric outside a neighborhood of 0 and have a small radially symmetric bump at the origin. If the bump is chosen properly, then dimfK.0/g D 1 and dimfKP g D 3 for P away from the origin. erefore, there are local Killing vector ﬁelds in this instance which do not extend to global Killing vector ﬁelds. e hypothesis that M is simply connected is needed as well. Give T 2 the ﬂat Riemannian metric. e rotation of R2 about 0 generates a Killing vector ﬁeld which restricts to a local Killing vector ﬁeld on T 2 which does not extend to a global Killing vector ﬁeld.

7.4

HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

If G is a compact Lie group which acts on a manifold M , then M admits a G invariant Riemannian metric. If the action is transitive, then M is compact. Lemma 7.12

Proof. Let be an arbitrary Riemannian metric on M . By Lemma 6.12, we can ﬁnd a bi-invariant measure j dvol j on G of total mass 1. If X; Y 2 TP M , let Z .x; y/ WD .g x; g y/j dvol j.g/ : G

is provides the required invariant Riemannian metric on M . If the action is transitive, then the map g ! g P is surjective and M is compact. t u Lemma 7.13

Every homogeneous Riemannian manifold is geodesically complete.

Proof. Let G be a Lie group which acts transitively and by isometries on a Riemannian manifold .M; g/. Let P 2 M . Choose > 0 so that every unit speed geodesic starting at P extends for time at least . Let W Œ0; T ! M be a unit speed geodesic with .0/ D P . Choose g 2 G so Lg P D .T /. Since Lg is an isometry, every unit speed geodesic from gP extends for time . erefore, we may extend to time Œ0; T C and hence, by induction, to time Œ0; 1/. t u

7.4.1 HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS. Since compact metric spaces are complete metric spaces, the Hopf–Rinow eorem (eorem 3.15 in Book I) shows that every compact Riemannian manifold is geodesically complete. In Section 7.4.2, we will exhibit a Lorentzian metric on the 2-torus that is not geodesically complete, and, in Section 7.4.3, we will exhibit an incomplete homogeneous Lorentzian metric. Consequently, a bit of care must be taken. ere is, however, an analog of Lemma 7.13 in the pseudo-Riemannian setting due to Marsden [39].

7.4. HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

101

Let .M; g/ be a compact homogeneous pseudo-Riemannian manifold. en .M; g/ is geodesically complete. eorem 7.14

Proof. Let be a geodesic deﬁned on a maximal domain Œ0; T /. We wish to show that T D 1. Suppose to the contrary that T < 1. Let tn ! T and let Qn WD .tn /. Since M is compact, by passing to a subsequence, we may assume that Qn ! Q for some point Q of M . By Lemma 7.10, there exist Killing vector ﬁelds fX1 ; : : : ; Xm g so fX1 .Q/; : : : ; Xm .Q/g form a basis for TQ M and hence form a frame for TM near Q. Expand .s/ P D ai .s/Xi ..s//. e matrix ij WD g.Xi ; Xj / is a non-singular symmetric matrix near Q. By Lemma 7.8, g..s/; P Xi ..s/// is independent of s . i i ij is implies gij ..s//a .s/ D cj or equivalently a .s/ D g . .s//cj . Since g ij is continuous near Q, g ij and hence ai .s/ is bounded near Q. Introduce an auxiliary Riemannian inner product .; /e near Q. We then have .; P / P e is uniformly bounded near Q. Since the tn ! T and Qn ! Q, this means that in fact .s/ stays near Q for tn < s < T . Consequently, .s/ tends to Q as s ! T . is implies that the maximal domain is not Œ0; T / and completes the proof. t u

7.4.2 AN INCOMPLETE LORENTZIAN METRIC ON THE TORUS. Every compact Riemannian manifold is geodesically complete. is can fail in the higher signature setting. e following example is due to Meneghini [43]. Let .N; gN / WD .R2 f0g; gN / where gN WD .x 2 C y 2 /

1

dx ı dy

has signature .1; 1/. Let m.x; y/ D .2x; 2y/. We then have m gN D ..m x/2 C .m y/2 /

1

d.m x/ ı d.m y/ D .4x 2 C 4y 2 /

1

4dx ı dy D gN :

is shows that m is an isometry which generates a cyclic subgroup which acts properly and discontinuously on N . Let M WD N=Z be the quotient by this subgroup; M is a compact manifold which is diﬀeomorphic to the torus S 1 S 1 and gN descends to deﬁne a Lorentzian metric gM on M . One uses the Koszul formula (see eorem 3.7 and Equation (3.3.a) in Book I) to compute: [email protected] @x ; @x / D 0;

[email protected] @x ; @y / D

2x ; .x 2 C y 2 /2

[email protected] @x D

2x @x : x2 C y2

Let .t/ D . 11 t ; 0/. We compute: r P P D fxR

2x

1

xP [email protected] P x D f2.1

t/

3

2.1

t /.1

t/

4

[email protected] D 0 :

erefore, is a geodesic with maximal domain . 1; 1/. Consequently, .N; gN / is geodesically incomplete so .M; gM / is geodesically incomplete as well. 7.4.3 AN INCOMPLETE HOMOGENEOUS LORENTZIAN METRIC. e following example is motivated by work of Guediri [24] and of Guediri and Lafontaine [25]. Let M be

102

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Minkowski space; M WD .R2 ; dx ı dy/. Let G be the ax C b group of Lemma 6.23. Deﬁne a smooth map T from G to I.M/ by setting T.˛;ˇ / .x; y/ WD .˛

1

x; ˛y C ˇ/ :

We verify that T is a group homomorphism by computing: T.˛;ˇ / ı T. ;ı/ W .x; y/ ! T.˛;ˇ / .

1

x; y C ı/ D .˛ 1 1 x; ˛ y C ˛ı C ˇ/ D T.˛ ;˛ıCˇ / .x; y/ D T.˛;ˇ /. ;ı/ .x; y/ :

Let M WD RC R and let M WD .M; dx ı dy/. en G acts transitively on M so M is a homogeneous Lorentzian manifold. e geodesics of this geometry are straight lines. rough any point of M there is exactly one straight line which is totally contained in M ; consequently, M is a geodesically incomplete. Since the map g ! g .1; 0/ is G -equivariant diﬀeomorphism from G to M , T .dx ı dy/ is a geodesically incomplete left-invariant Lorentzian metric on the ax C b group.

7.5

LOCAL SYMMETRIC SPACES

ere is both a local and a global aspect to the theory of symmetric spaces in the pseudoRiemannian context. We begin by presenting results of Cartan [9, 10]; the proof of these results uses Jacobi vector ﬁelds to examine the local geometry. Let R be the curvature operator of a pseudo-Riemannian manifold .M; g/. We deﬁne the covariant derivative of the curvature operator rX R and the covariant derivative of the curvature tensor rR by setting: f.rX R/.Y; Z/gW

WD rX fR.Y; Z/W g R.rX Y; Z/W R.Y; rX Z/W R.Y; Z/rX W; rR.Y; Z; W; U I X/ WD g.rX R.Y; Z/W; U / D X.R.Y; Z; W; U // R.rX Y; Z; W; U / R.Y; rX Z; W; U / R.Y; Z; rX W; U / R.Y; Z; W; rX U / :

Both rR and r R are tensors. If .M; g/ has constant sectional curvature, then r R D 0 as we shall see presently in Example 7.6.1. Consequently, the following result can be regarded as a generalization of Lemma 3.19 in Book I. Lemma 7.15 Let M and MQ be pseudo-Riemannian manifolds with curvature operators R and Q , respectively, with r R D 0 and rQ R Q D 0. Let P 2 M and let PQ 2 MQ . Suppose there exists a R Q .PQ / D R.P /. en the map linear isometry from TP M to T Q MQ so that R P

WD expPQ ı

ı logP

is a local isometry from a neighborhood of P in M to a neighborhood of PQ in MQ . Furthermore, if M and MQ are simply connected and geodesically complete, then there exists a unique global isometry from M onto MQ so that .P / D PQ and .P / D .

7.5. LOCAL SYMMETRIC SPACES 1

103

m

Proof. Fix a basis fe1 ; : : : ; em g for TP M and let xE D expP .x e1 C C x em / be geodesic coordinates near P . Fix v 2 Rm . e straight line .s/ WD s v in the direction of v through the origin is a geodesic in M . Let fE1 .s/; : : : ; Em .s/g be the parallel frame ﬁeld along with initial condition Ei .0/ D ei . Since r R D 0, the components of the curvature tensor relative to the frame fEi g are constant along : @s R.Ei ; Ej ; Ek ; E` / D .rP R/.Ei ; Ej ; Ek ; E` / C R.rP Ei ; Ej ; Ek ; E` / CR.Ei ; rP Ej ; Ek ; E` / C R.Ei ; Ej ; rP Ek ; E` / C R.Ei ; Ej ; Ek ; rP E` / D 0 :

If Y is a vector ﬁeld along , let J .P / W Y W! R.Y; / P P be the Jacobi operator; Y is said to R be a Jacobi vector ﬁeld if Y C R.Y; / P P D 0. Let T .s; t / WD s.v C t w/ be a geodesic spray. By Lemma 3.16 in Book I, the associated variational vector Y .s/ D @ t T .s; t /j tD0 D sw is a Jacobi vector ﬁeld along . Since the components of the curvature tensor are constant relative to the frame Ei , Y satisﬁes a constant coeﬃcient ordinary diﬀerential equation relative to the moving frame fEi .s/g with initial condition Y.0/ D 0 and YP .0/ D w . Consequently, the components of Y are completely determined by Y.0/ D 0 and YP .0/ D w . erefore, g.sw; sw/. .s// is determined by the data fs; g.ei ; ej /; .w; ei /; Rij k` .0/g where .w; ei / is the usual Euclidean inner product. Let eQi D ei . We perform the same construction on MQ where Q D . As .expP .s.v C t w/// D expPQ .s. v C t w// ;

we have .Y / D YQ . Since g.Y.s/; Y.s//. .s// D g. Q YQ .s/; YQ .s//..s// Q , is an isometry away from 0; at 0, is an isometry by construction. Assume additionally that .M; g/ and .MQ ; g/ Q are geodesically complete and simply connected. Let .t/ WD expP .tv/ and let .t Q / WD expPQ .t v/ be geodesics on M and on MQ , respectively. Fix 0 < T < 1. Since .Œ0; T / and Q .Œ0; T / are compact, we can cover .Œ0; T / and .Œ0; Q T / by a ﬁnite number of open sets where the inverse of the exponential map is a diﬀeomorphism. We can then recursively apply the local construction described above to extend as a local isometry along .s/ so ı D Q . We can join any two points of M by broken geodesics and extend along a path of broken geodesics. e assumption that M is simply connected then implies there is no problem with holonomy and deﬁnes W M ! MQ so that .g/ Q D g . We 1 apply a similar construction to to construct 1 and see is a global isometry with the desired properties. By Lemma 7.4, is uniquely determined by the conditions .P / D PQ and .P / D . t u We remark that the assumption that M and MQ are simply connected in Lemma 7.15 is Q WD .R2 ; dx 2 C dy 2 /, then M and M Q are comnecessary. If M WD .T 2 ; dx 2 C dy 2 / and if M plete ﬂat Riemannian manifolds which are locally isometric; there is, however, no global isometry Q . Similarly, the assumption that M and M Q are complete in Lemma 7.15 is necfrom M onto M 2 2 Q WD ..0; 1/; dx /, then M and M Q are simply connected essary. If M WD ..0; 2/; dx / and if M ﬂat Riemannian manifolds which are locally isometric. But there is no global isometry from M Q. onto M

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7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

7.5.1 THE GEODESIC INVOLUTION. Let .M; g/ be a pseudo-Riemannian manifold. e geodesic involution SP about a point P of M is deﬁned by setting SP .Q/ WD expP f logP .Q/g :

(7.5.a)

e map SP is a local diﬀeomorphism deﬁned near P which is characterized by the property that .SP /.t/ D . t/ for any geodesic through P . Lemma 7.16

Let .M; g/ be a connected pseudo-Riemannian manifold.

1. e following assertions are equivalent and if any is satisﬁed, then .M; g/ is said to be a local symmetric space: (a) e geodesic symmetry SP is a local isometry deﬁned near P for all P 2 M . (b) r R D 0. is means that the curvature operator is parallel. (c) If X , Y , and Z are parallel vector ﬁelds along a curve , then R.X; Y /Z is a parallel vector ﬁeld along . 2. If .M; g/ is a local symmetric space, then .M; g/ is locally homogeneous. Proof. Let xE be a system of geodesic coordinates on M centered at P . en SP .x/ E D xE . Let Ai D @xi be coordinate vector ﬁelds; .SP / .Ai / D Ai . If SP is a local isometry, then we have that SP .rR/ D rR. Consequently, . 1/5 rA1 R.A2 ; A3 ; A4 ; A5 /.0/ D rR A1 . A2 ; A3 ; A4 ; A5 /.0/ D rA1 R.A2 ; A3 ; A4 ; A5 I A1 /.0/ :

is implies that rR.0/ D 0 and shows that Assertion 1-a implies Assertion 1-b. Conversely, if Assertion 1-b holds, then we may apply Lemma 7.15 with M D MQ , P D PQ , and D Id to see that SP is a local isometry and show that Assertion 1-a holds. Let fei g be a parallel frame along a curve . e equivalence of Assertion 1-b and Assertion 1-c follows from the identity .r P R/.ei ; ej ; ek ; e` / D R.e P i ; ej ; ek ; e` /. Assume that SP is a local isometry for all P 2 M . Let be a curve from a point Q1 to a point Q2 of M . Since rangef g is compact, we may cover rangef g by a ﬁnite number of open sets so that any two points in one of the open sets can be joined by unique shortest geodesic and so that the geodesic involution about the midpoint of such a geodesic is a local isometry interchanging the two end points. Composing such geodesic involutions then yields a local isometry interchanging Q1 and Q2 . t u

7.6

THE GLOBAL GEOMETRY OF SYMMETRIC SPACES

We say that a connected pseudo-Riemannian manifold .M; g/ is a symmetric space if the geodesic involution SP of Equation (7.5.a) extends for any P 2 M to a global isometry of M . e following is an immediate consequence of Lemma 7.15.

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105

Lemma 7.17 If .M; g/ is a simply connected geodesically complete local symmetric space, then .M; g/ is a symmetric space.

e assumption that .M; g/ is simply connected is essential in Lemma 7.17. Let the group of nth roots of unity Zn act by complex multiplication without ﬁxed points isometrically on S 2k 1 C k . Give the quotient manifold L.n; k/ WD S 2k 1 =Zn the induced metric so that the covering projection from S 2k 1 to L.nI k/ is a local isometry; L.n; k/ is an example of a lens space. en L.nI k/ is locally symmetric and geodesically complete. If P D .1; 0; : : : ; 0/, then SP .x/ E D .x 1 ; x 2 ; : : : ; x 2k /. Since SP does not commute with the action of Zn for n 3, L.nI k/ is not globally symmetric. Lemma 7.18 If .M; g/ is a symmetric space, then the map ˚ W .P; Q/ ! SP .Q/ is a smooth map from M M to M .

Proof. Let B .Q/ be the open ball of radius about Q relative to some distance function on M deﬁning the topology. Let P; PQ 2 M . We must show that the map ˚ is smooth near .P; PQ /. Let W Œ0; T ! M be a curve from P to PQ in M . Choose > 0 so B2 .R/ domainflogR g for any R 2 rangefg, i.e., so expR is a diﬀeomorphism from logP .B2 .R// to B2 .R/. Choose ı > 0 so that if js tj < ı , then d..s/; .t// < and so that T D nı for some n 2 N . Let Rj WD .j ı/ for 1 j n. If Q 2 B .P / and if R 2 B .P /, then R 2 B2 .Q/ and ˚.Q; R/ D SQ .R/ D expQ . logQ .R// :

Consequently, ˚ is smooth on B .P / B .P /. We have P D R0 . Choose 0 j n maximal so ˚ is smooth on B .P / B .Rj /. Suppose j < n; we argue for a contradiction. Let Q belong to B .P /. As SQ is a global isometry of M , SQ commutes with the exponential map. Let R belong to B2 .Rj /. en ˚.Q; R/ D SQ .R/ D SQ .expRj .logRj .R/// D expSQ .Rj / .SQ .Rj / .logRj .R///

so ˚ is smooth on B .P / B2 .Rj /. Since d.Rj C1 ; Rj / < , B .Rj C1 / B2 .Rj / and ˚ is smooth on B .P / B .Rj C1 /. is contradicts the maximality of j . Consequently, j D n and, since R D Rn , ˚ is smooth on B .P / B .R/. t u Let .M; g/ be a symmetric space. If is a non-constant geodesic, then the associated transvection is deﬁned by setting: t D t; WD S . 1 t / S.0/ 2 I .M; g/ : 2

(7.6.a)

e following result provides a partial converse to Lemma 7.17 by showing that any symmetric space is geodesically complete.

106

7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Lemma 7.19 Let .M; g/ be a symmetric space. Let t WD S. 1 t/ S.0/ be the transvection of 2 a geodesic . en t ..s// D .t C s/, .M; g/ is geodesically complete, and t is deﬁned for all t 2 R. Either is injective or is simply periodic. If X is a parallel vector ﬁeld along , then . t / X D X . e map . t / ..s// W T.s/ M ! T .t Cs/ .M / is given by parallel translation along . Finally, f t g t 2R is a smooth one-parameter group of isometries in the group of isometries I .M; g/.

Proof. Since S.t / .s/ D .2t s/ for t and s small, the transvection t of Equation (7.6.a) satisﬁes: t .s/ D S. 1 t/ S.0/ .s/ D S . 1 t / . s/ D .t C s/ : 2

2

We may use the transvections t to translate the geodesic and to extend (which a-priori is only deﬁned on . ; / for some > 0) to . 1; 1/. Consequently, .M; g/ is geodesically complete. If .0/ D .b/ for some b , then .t / D t .0/ D t .b/ D .b C t/ for any t so is simply periodic. Let X be a parallel vector ﬁeld along . Since S .t / is an isometry, it preserves parallel vector ﬁelds. Since .S.t / / ..t // D 1 on T .t / M , S.t / X D X for any t . erefore, . t / X D X as we have changed the sign twice. Furthermore, . t / ..s//X. .s// D X. .s C t //

so the action of . t / mapping T.s/ to T.t / is given by parallel translation. By Lemma 7.4, an isometry is determined by its value and its derivative at a single point. We have t s .0/ D t f.s/g D .s C t/ :

Furthermore, ﬁrst parallel translating from .0/ to .s/ and then parallel translating from .s/ to .t/ is parallel translating from .0/ to .s C t /. Consequently, t s D tCs and t is a 1-parameter family. By Lemma 7.18, the map .t; Q/ ! S . 1 t / S .0/ is smooth in .t; Q/. Conse2 quently, by eorem 7.6, t is a smooth 1-parameter family of the group of isometries I .M; g/. u t eorem 7.20 Let .M; g/ be a simply connected symmetric space. en any local Killing vector ﬁeld on M extends to a global Killing vector ﬁeld on M , the vector space of Killing vectors is the Lie algebra of I .M; g/, and .M; g/ is a homogeneous space.

Proof. By Lemma 7.19, .M; g/ is geodesically complete. We adopt the notation of eorem 7.11. Let W D TM ˚ .T M ˝ TM /. If X is a local Killing vector ﬁeld, let SX .P / WD X.P / ˚ rX.P / 2 WP

and let KP D [X SP .X/ be a linear subspace of WP . By Lemma 7.16, .M; g/ is locally homogeneous. us, dimfKP g is constant on M . By eorem 7.11, since .M; g/ is simply connected and geodesically complete, every local Killing vector ﬁeld on M extends to a global Killing vector ﬁeld.

7.6. THE GLOBAL GEOMETRY OF SYMMETRIC SPACES

107

By Lemma 7.9, since .M; g/ is simply connected and geodesically complete, every global Killing vector ﬁeld deﬁnes a 1-parameter family of isometries and the Lie algebra of global Killing vector ﬁelds is the Lie algebra of the isometry group I .M; g/. Fix P 2 M . e transvections deﬁned by the geodesics through P deﬁne m linearly independent Killing vector ﬁelds at P . Consequently, by Lemma 7.10, .M; g/ is homogeneous. t u

Let .M; g/ be a pseudo-Riemannian manifold. Let N be a connected smooth connected submanifold of M so that gjN is non-degenerate and so that .N; gjN / is totally geodesic. If .M; g/ is a local symmetric space, then .N; gjN / is a local symmetric space. If .M; g/ is a symmetric space, then .N; gjN / is a symmetric space. eorem 7.21

Proof. Assume that .M; g/ is a local symmetric space and that N is a totally geodesic nondegenerate submanifold of M . Let P 2 N . Since N is totally geodesic, the geodesic symmetry SP of M preserves N and hence N is a local symmetric space. If M is a symmetric space, then SP is globally deﬁned. Let NQ WD fQ 2 N W SP .Q/ 2 N and .SP / .Q/ W TQ N ! TSP .Q/ .N /g :

en NQ is a closed set. As N is totally geodesic, it follows that NQ is open. Since N is connected, NQ D N and the restriction of SP to N provides the requisite geodesic involution on N . t u Let .M; g/ be a symmetric space. By eorem 7.20, the connected component of the isometry group I0 .M; g/ acts transitively on M . Let H D HP I0 .M; g/ be the isotropy subgroup of the connected component of the identity of the isometry group of .M; g/. By eorem 7.2, the natural map ! P is a diﬀeomorphism from I0 .M; g/=H to M . Note that the geodesic involution SP belongs to I .M; g/ but that SP need not belong to I0 .M; g/. If we take .M; g/ D .T 2 ; dx 2 C dy 2 /, then SP acts non-trivially on the de Rham cohomology of the torus and hence Sp is not in the identity component of the isometry group. On the other hand, if we take .M; g/ D .R2 ; dx 2 C dy 2 / and let T .P / be rotation through an angle about some point P 2 R2 , t .P / is a smooth 1-parameter family of isometries about P with D SP and SP 2 I0 .M; g/. Deﬁne an involution D P of I .M; g/ by setting . / WD SP ı ı SP :

(7.6.b)

Since I0 .M; g/ is a normal subgroup of I .M; g/, it is preserved by . Let G .M; g/ WD f 2 I0 .M; g/ W . / D g :

is Lie group need not be connected. Let G0 .M; g/ be the connected component of the identity of the Lie group G ; in Example 7.6.1 we exhibit .G; / so that G ¤ G0 .

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7. HOMOGENEOUS SPACES AND SYMMETRIC SPACES

Lemma 7.22

If .M; g/ is a symmetric space, then G0 .M; g/ H.M; g/ G .M; g/.

Proof. If 2 H , let WD 1 SP SP . Since SP .P / D P and .P / D P , .P / D P . We use the chain rule to see that .P / D . .P / 1 /. Id/. .P //. Id/ D Id. Consequently, by Lemma 7.4, D Id and sP sP D . is shows H G :

We show next that G0 H . Since G0 is connected, G0 is generated by the 1-parameter subgroups of G0 and it suﬃces to show that exp.t / 2 H for any in the Lie algebra of G0 . We have SP exp.t/SP 1 D exp.tSP SP / D exp.t Ad.SP // : is shows that the Lie algebra of G0 consists of the ﬁxed points of the involution under the adjoint action. Consequently, if 2 g.G /, then SP exp.t/ D exp.t/SP . erefore, SP exp.t/P D P . Since P is an isolated ﬁxed point of SP , this implies exp.t/P D P and hence exp.t/ 2 H as desired. t u We now pass to the algebraic level. eorem 7.23 Let be a group homomorphism of a connected Lie group G with 2 D Id. Let G WD f 2 G W ./ D g; G is a closed Lie subgroup of G . Let G0 be the connected component of the identity in G . Let H be a closed subgroup of G with G0 H G . en every G -invariant pseudo-Riemannian metric on M D G=H gives M the structure of a symmetric space. Let be the natural projection from G to G=H . en . .g// D SP .g/ where SP is the geodesic involution based at P D .Id/.

Proof. Let M D G=H be given a G -invariant pseudo-Riemannian metric. Let P D .Id/ be the base point. Since the involution ﬁxes H , we have .g1 H / D .g1 / .H / D .g1 /H :

If .g1 / D .g2 /, then g1 H D g2 H and, thus, . .g1 // D . .g2 //. Consequently, SP .g/ WD ..g// is well-deﬁned. e existence of local sections to the submersion implies that SP is smooth. Clearly, SP .P / D P . Since 2 D Id, SP2 D Id. us, SP is a smooth involution of M . Let g and h be the Lie algebras of G and H , respectively. We then have that g is the direct sum of h and the subspace m D fX 2 g W d .X / D Xg. We argue as follows to see this. If X 2 g, decompose X D Xh C Xm , where Xh D 12 .X C d .X // and Xm D 12 .X d .X //. Since is involutive, so is d , and, therefore, d .Xh / D Xh and d .Xm / D Xm . It follows that g D h C m, which is a direct sum since h \ m D 0. Next, note that SP .P / D P and if y 2 TP M the previous argument implies that there exists Y 2 g such that d .Y / D Y and d .Y / D y . We show that dSP D Id by computing:

7.6. THE GLOBAL GEOMETRY OF SYMMETRIC SPACES

109

y.

dSP .y/ D dSP .d .Y // D d .d .Y // D d . Y / D

Finally, let h ; i be a G -invariant metric tensor on M D G=H and associated to each element g in G consider the translation g W G=H ! G=H given by g .g1 H / D gg1 H . For a 2 G , we have SP .g ..a/// D SP ..ga// D . .ga// D . .g/ .a// D .g/ .. .a/// D .g/ .SP ..a/// ;

so .g/ D SP g SP . As a consequence, if v 2 Tg M and vP D d g

1

.v/ 2 TP M , we get

hdSP .v/; dSP .v/i D hdSP d g .vP /; dSP d g .vP /i D hd .g/ dSP .vP /; d .g/ dSP .vP /i D hdSP .vP /; dSP .vP / D h vP ; vP i D hv; vi

is shows that the map SP is an isometry with respect to h ; i. If a homogeneous space has a global symmetry at P , then SP 1 is a global symmetry at any point .P /. is completes the proof. t u By Lemma 7.22, every symmetric space arises from the construction of eorem 7.23. Consequently, eorem 7.23 reduces the study of symmetric spaces to a group theoretic problem. We refer to Helgason [30] for a further discussion in this regard; this is a vast ﬁeld. Let G be a compact Lie group and let 2 G be involutive. Suppose that H is a closed subgroup of G with .G0 /0 H .G0 / . Set M WD G0 =H . By Lemma 7.12, there exists a Riemannian metric ds 2 on M so G acts by isometries; .M; ds 2 / is then a symmetric space. 7.6.1 EXAMPLE. Let G D SO.m/ and let W SO.m/ ! SO.m/ be given by: .A/ WD

Idk 0

0 Idm

!

k

A

Idk 0

0 Idm

!

1

:

k

en G D fO.k/ O.m k/g \ SO.m/ and G0 D SO.k/ SO.m k/g; there is a natural double cover Z2 ! G ! G0 . Taking H D G yields the Grassmann manifold Grk .Rm / of k m planes in Rm ; taking H D G0 yields the Grassmann manifold GrC k .R / of oriented k planes in m m Rm . e double cover Z2 ! GrC k .R / ! Grk .R / arises by forgetting the orientation. Taking m 1 k D 1 yields the sphere S and real projective space RP m 1 , respectively. 7.6.2 EXAMPLE. A Lie group G is a symmetric space determined by the symmetric pair .G G; G /, where G is the diagonal subgroup G D f.g; g/ 2 G G W g 2 Gg and where the involution interchanges the two factors, i.e., .g1 ; g2 / D .g2 ; g1 /; we may take any leftinvariant pseudo-Riemannian metric on G .

111

CHAPTER

8

Other Cohomology eories In Chapter 8, we show that de Rham cohomology is isomorphic to singular cohomology, PL cohomology, and sheaf cohomology. We begin in Section 8.1 by presenting some standard results in homological algebra that we used to complete the proof of eorem 5.2 by establishing the Mayer–Vietoris sequence in de Rham cohomology and by showing that de Rham cohomology is a homotopy functor. In Section 8.2, we discuss simplicial theory. In Section 8.3, we discuss singular (i.e., topological or simply TP) cohomology HTP ./ and HTP ./. In Section 8.4, we treat sheaf cohomology.

8.1

HOMOLOGICAL ALGEBRA

We begin with some basic deﬁnitions. 8.1.1 THE LANGUAGE OF CATEGORY THEORY. e reader may safely bypass this material as the terminology involved will be, for the most part, clear by context subsequently; it is presented for the sake of completeness. Let C be a category. We shall work in the category of simplicial complexes (the PL category), the category of topological spaces (the TP category), and the category of smooth manifolds (the C 1 category). Let H be a contravariant functor from the category C to the category of real or complex vector spaces. If X is an element of C, let H.X/ be the associated vector space. If f W X ! Y is a morphism in the category, then we have an associated map f W H.Y / ! H.X/ so that

Id D Id

and .f ı g/ D g ı f :

e morphisms in the category of smooth manifolds are the smooth maps, the morphisms in the category of topological spaces are the continuous maps, and the morphisms in the simplicial category are the simplicial maps. As discussed in Chapter 5, de Rham cohomology is a functor which is deﬁned on the smooth category. Simplicial cohomology (see Section 8.2) is deﬁned on the PL-category, singular cohomology (see Section 8.3) is deﬁned on the topological category, and sheaf cohomology (see Section 8.4) is deﬁned on the topological category. We have natural inclusions of the simplicial and smooth categories in the topological category. If H1 is a functor on a category C1 , if H2 is a functor on a category C2 , and if i W C1 ! C2 is a map between categories, then a natural transformation of functors is a linear map X from H1 .X/ to H2 .i.X // for each element X of C1 so that if f W X ! Y is a morphism in C1 , then we have a commutative diagram:

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8. OTHER COHOMOLOGY THEORIES

H1 .Y / # f1 H1 .X /

Y

! H2 .i.Y // ı # f2 . X ! H2 .i.X //

8.1.2 CHAIN COMPLEXES AND COCHAIN COMPLEXES. A cochain complex A is a collection of real vector spaces Ai together with linear maps diA W Ai ! AiC1 where the Ai are real (or possibly complex) vector spaces and where the diA are linear maps so that diA diA 1 D 0; it may be represented by the diagram: dpA

d0A

0 ! A0 !A1 ! ! Ap

1

dpA

1

!Ap !ApC1 :

e cohomology of the cochain complex A is deﬁned to be: H i .A/ WD

kerfdiA W Ai ! AiC1 g : rangefdiA 1 W Ai 1 ! Ai g

If a 2 kerfdiA g, we let Œa 2 H i .A/ denote the corresponding element in cohomology. A cochain map T from a cochain complex A D .Ai ; diA / to a cochain complex B D .Bi ; diB / is a collection of linear maps T D .Ti W Ai ! Bi / so diB Ti D TiC1 diA . We represent this as: 0 !

A0 # T0

0 !

B0

d0A

! A1 ı # T1

! ! Ai ::: # Ti

!

! !

d0B

B1

Bi

diA

! ı

diB

AiC1 # TiC1 :

! BiC1

Let S and T be cochain maps from a cochain complex A to a cochain complex B . We say that R D fRi W Ai ! Bi 1 g is a cochain homotopy between S and T if diB 1 Ri C RiC1 diA D Ti Si . ere is always a question about notation; we will write the indices on the vector spaces down, the maps will go to the right or down, and the indices on the cohomology groups will be up. When no confusion is likely to result, we suppress superscripts and subscripts and set d D diA and T D Ti . Lemma 8.1 Let T be a cochain map from a cochain complex A to a cochain complex B . en the map T W Œa ! ŒT a is a well-deﬁned map in cohomology from H p .A/ to H p .B /. If T and S are cochain homotopic, then T D S .

Proof. Let 2 H p .A/. Find ap 2 Ap so Œap D . Since dap D 0, d T ap D T dap D 0 so ŒT ap is well-deﬁned in H p .B /. If ŒaQp D is another possible representative of , then there exists ap 1 2 Ap 1 so dap 1 D ap aQp . Consequently, T .ap aQp / D T dap 1 D d T ap 1 . is shows that ŒT ap D ŒT aQp in H p .B / and T is well-deﬁned in cohomology. Let T and S be chain homotopic. As dap D 0, ŒSap

T ap D ŒdRap C Rdap D ŒdRap C 0 D 0

and, consequently, S D T as maps from H p .A/ to H p .B /.

t u

8.1. HOMOLOGICAL ALGEBRA ˛

113

ˇ

8.1.3 SHORT EXACT SEQUENCES. We say that 0 ! A !B !C ! 0 is a short exact sequence of vector spaces if the maps ˛ and ˇ are linear, if ˛ is injective, if ˇ is surjective, and if kerfˇg D rangef˛g. We say that a cochain complex A is a long exact sequence if H i .A/ D 0 for all i , i.e., if kerfdiA g D rangefdiA 1 g. Massey [40] (p. 185) states: “Apparently exact sequences were introduced into algebraic topology by Witold Hurewicz in 1941” (see [34]). We now discuss the combinatorial Laplacian. e following is a useful observation that is the extension of eorem 5.13 to the setting at hand.

Let A be a cochain complex where each Ai is a ﬁnite-dimensional vector space. Put a Hermitian inner product h; i on each Ai and let ıi W Ai ! Ai 1 be the adjoint map. Let WD ıd C d ı be the associated Laplacian. en: Lemma 8.2

1. kerfg D kerfd g \ kerfıg, Ai D kerfi g ˚ rangefdi 1 g ˚ rangefıi g is a direct orthogonal decomposition, and d is an isomorphism from rangefıg to rangefd g. Finally, the map which sends a to Œa deﬁnes an isomorphism from kerfp g to H p .A/. P P 2. i . 1/i dimfAi g D i dimfH i .A/g. In particular, if A is a long exact sequence, then P i i i . 1/ dimfH .A/g D 0. Proof. We suppress indices for the moment. If a D 0, then 0 D ha; ai D hıda; ai C hd ıa; ai D hda; dai C hıa; ıai D kdak2 C kıak2 :

is shows kerfg D kerfd g \ kerfıg. As is self-adjoint, we have an orthogonal direct sum decomposition A D kerfg ˚ rangefg D kerfg ˚ frangefd ıg C rangefıd gg :

(8.1.a)

Since hda; ı ai Q D hdda; ai Q D 0, rangefd g is perpendicular to rangefıg. us, the decomposition of Equation (8.1.a) is an orthogonal direct sum decomposition. If d ıa2 D 0, then 0 D hd ıa2 ; a2 i D hıa2 ; ıa2 i

so ıa2 D 0. is implies d is an injective map from rangefıg to rangefd g. Similarly, we have ı is an injective map from rangefd g to rangefıg. us, dimfrangefıgg D dimfrangefd gg and d is an isomorphism from rangefıg to rangefd g. Let a 2 kerfg. en a 2 kerfd g so Œa is an element of cohomology. If Œa D 0, then a is in the range of d . But kerfg ? rangefd g. erefore, a D 0 so the map a ! Œa is an injective map from kerfg to cohomology. Conversely, suppose da D 0. Decompose a D a0 C da1 C ıa2 where a0 2 kerfg. en 0 D d ıa2 . is implies that ıa2 D 0 and that a D a0 C da1 . Consequently, Œa D Œa0 . is completes the proof of Assertion 1. Let Rid WD rangefdi 1 g and Riı 1 WD rangefıi g. Because di 1 is an isomorphism from rangefıi g Ai 1 to rangefdi 1 g Ai , we have dimfRid g D dimfRiı 1 g. Consequently,

114

8. OTHER COHOMOLOGY THEORIES

P

1/i dimfH i .C /g P P D i . 1/i dimfkerfi gg C i . 1/i fdimfRid g dimfRiı 1 gg P P P D i . 1/i dimfkerfi gg C i . 1/i dimfRid g C i . 1/i dimfRiı g P D i . 1/i dimfAi g . i.

If A is a long exact sequence, then the cohomology is trivial so the sum vanishes.

t u

8.1.4 SHORT EXACT SEQUENCE OF CO-CHAIN COMPLEXES. A short exact sequence ˇ ˛ of cochain complexes 0 ! A !B !C ! 0 is a pair of cochain maps ˛ from A to B and ˇ from ˇ ˛ B to C so that the column maps form a short exact sequence 0 ! Ap !Bp !Cp ! 0. Consequently, we have a commutative diagram: 0 # A0

0!

# ˛0 0!

B0 # ˇ0

0!

0 # A1

d0A

!

d0B

ı

# ˛1

ı

# ˇ1

!

d0C

B1

0 # Ai

# ˛i

Bi

# ˇi

diA

!

0 #

AiC1

diB

# ˛iC1

ı

# ˇiC1

ı

!

diC

BiC1

:

C0 ! C1 Ci ! CiC1 # # # # 0 0 0 0 e following result is due to the French mathematician Élie Cartan and the Polish mathematician Samuel Eilenberg [12] (p. 40); it is often called the Snake Lemma.

E. Cartan (1869–1951)

S. Eilenberg (1913–1998)

ˇ

˛

Let 0 ! A !B !C ! 0 be a short exact sequence of cochain complexes. ere exists a natural map v W H p 1 .C / ! H p .A/, which is called the connecting homomorphism, giving rise to a long exact sequence in cohomology:

Lemma 8.3

˛

ˇ

v

˛

ˇ

v

0 ! H 0 .A/ !H 0 .B / !H 0 .C / !H 1 .A/ !H 1 .B / !H 1 .C / !H 2 .A/ ! :

8.1. HOMOLOGICAL ALGEBRA

115

e connecting homomorphism v is natural in the sense that a commutative diagram of short exact sequences of cochain complexes 0 ! A # A 0 ! AQ

˛

! B ı # B ˛Q ! BQ

ˇ

! C ı # C ˇQ ! CQ

! 0 ! 0

gives rise to a commutative diagram of long exact sequences: ˇ

˛

ˇ

˛

v

v

0 ! H 0 .A/ !H 0 .B / !H 0 .C / !H 1 .A/ ! H 1 .B / ! H 1 .C / ! H 2 .A/ # .A / ı # .B / ı # .C / ı # .A / ı # .B / ı # .C / ı # .A / : ˇQ ˇQ Q Q ˛Q ˛Q v v 0 ! H 0 .AQ / !H 0 .BQ / !H 0 .CQ/ !H 1 .AQ / ! H 1 .BQ / ! H 1 .CQ/ ! H 2 .AQ /

Proof. e proof is called diagram chasing by Massey [40] (p. 184) since it “… requires practically no cleverness or ingenuity. At each stage of the proof, there is only one possible ‘move’; one does not have to make any choices.” e connecting homomorphism v is constructed as follows. Let cp 1 2 Cp 1 satisfy dcp 1 D 0. Since ˇ is assumed to be surjective, we can ﬁnd bp 1 2 Bp 1 so that ˇbp 1 D cp 1 . en ˇdbp 1 D dˇbp 1 D dcp 1 D 0

so there is a unique ap 2 Ap so ˛ap D dbp 1 . We compute ˛dap D d˛ap D ddbp 1 D 0. Since ˛ is injective, dap D 0 so Œap is well-deﬁned in H p .A/. e picture may be drawn as follows: 0 # ap #˛ d ! bp ı #ˇ d ! 0

bp 1 #ˇ cp 1 # 0

Let bQp ˇ.bp 1

1

0 # d ! d˛p ı #˛ d ! 0 :

be another possible lift, i.e., ˇ bQp 1 D cp 1 . Choose aQp so that ˛ aQp D d bQp Qbp 1 / D 0, there exists ap 1 so bp 1 bQp 1 D ˛.ap 1 /. en ˛dap

1

D d˛ap

1

D dbp

1

d bQp

1

D ˛.ap

1.

As

aQp / :

Consequently, as ˛ is injective, ap aQp D dap 1 so Œap D ŒaQp in H p .A/. erefore, the map v which sends cp to Œap is a well-deﬁned map from kerfdpC 1 g to H p .A/. Next, suppose that

116

8. OTHER COHOMOLOGY THEORIES

cp 1 D dcp 2 . Find bp 2 so ˇbp 2 D cp 2 . en ˇdbp 2 D dˇbp 2 D dcp 2 D cp 1 and, consequently, dbp 2 will do as a lift of cp 1 . Since ddbp 2 D 0, we have ap D 0 in this instance and vcp 1 D 0. is shows v is a well-deﬁned map v W Hp

1

.C / ! H p .A/ :

We now turn to the sequence of Lemma 8.3. We begin by examining exactness at H p .B /. Clearly, ˇ ˛ Œa D Œˇ˛a D 0. Conversely, suppose dbp D 0 and ˇ Œbp D 0. ere then exists cp 1 so dcp 1 D ˇbp . Choose bp 1 so ˇbp 1 D cp 1 . en ˇbp D dcp 1 D dˇbp 1 and, consequently, ˇ.bp dbp 1 / D 0. us, there exists ap so bp dbp 1 D ˛ap . Since ˛dap D d˛ap D dbp D 0

and since ˛ is injective, dap D 0. Consequently, Œap 2 H p .A/ satisﬁes ˛ Œap D Œbp

dbp

1

D Œbp :

is shows kerfˇ g D rangef˛ g :

Next, we examine exactness at H p 1 .C /. Let Œcp 1 2 rangefˇ g. By adjusting cp 1 by an element of rangefd g, we may assume cp 1 D ˇbp 1 where dbp 1 D 0. Since dbp 1 D 0, the corresponding lift ap D 0 and, consequently, v.cp 1 / D 0. erefore, Œcp 1 2 kerfvg. Conversely, suppose dcp 1 D 0 and Œcp 1 2 kerfvg. Choose bp 1 so ˇbp 1 D cp 1 . Express dbp 1 D ˛ap . Since Œap D vŒcp 1 D 0, ap D dap 1 . Consequently, d.bp

Consequently, Œbp

1

˛ap

1

1

2 Hp

˛ap 1

1/

D dbp

.B / and ˇ Œbp

1

˛ap D 0 :

1

˛ap

1

D Œcp

1 .

is shows

kerfvg D rangef˛ g : Finally, we examine exactness at H p .A/. Let Œap 2 rangefvg. Let ˛ap D dbp 1 for ˇbp 1 D cp 1 . is implies ˛Œap D 0 so Œap 2 kerf˛ g. Conversely, suppose ˛ Œap D 0. We then have ˛ap D bp for bp D dbp 1 . Set cp 1 D ˇbp 1 . en dcp

So Œcp

1

2 Hp

1

1

D dˇbp

.C / and vŒcp

1

1

D Œap

D ˇdbp 1 .

1

D ˇbp D ˇ˛ap D 0 :

Consequently,

kerfˇ g D rangefvg : is shows the sequence is exact. Clearly, v is a natural map.

8.1.5 THE 5-LEMMA. We will use the following observation repeatedly to establish the equivalence of various cohomology theories. It is called the 5-Lemma. We refer to Eilenberg and Steenrod [18] (p. 16) for details.

8.1. HOMOLOGICAL ALGEBRA

Lemma 8.4

117

Suppose given a commutative diagram of exact sequences: A1 # ˛1 B1

d

! A2 ı # ˛2

d

!

B2

d

! A3 ı # ˛3

d

!

B3

d

! A4 ı # ˛4

d

!

B4

d

! A5 ı # ˛5 :

d

!

B5

Assume that ˛1 , ˛2 , ˛4 , and ˛5 are isomorphisms. en ˛3 is an isomorphism. Proof. Again the proof is a diagram chase. We shall ﬁrst show that ˛3 is surjective. Let b3 2 B3 . Since ˛4 is surjective, we may choose a4 2 A4 so ˛4 a4 D db3 : en ˛5 da4 D d˛4 a4 D d db3 D 0 :

As ˛5 is injective, da4 D 0. As the sequence is exact at A4 , we may choose a3 2 A3 so that da3 D a4 . Let bQ3 D b3 ˛3 a3 . One then has that: d bQ3 D db3

d˛3 a3 D db3

˛4 da3 D db3

˛4 a4 D 0 :

Because the sequence is exact at B3 , we may choose b2 2 B2 so that bQ3 D db2 . As ˛2 is surjective, we may choose a2 2 A2 so ˛2 a2 D b2 . We have ˛3 da2 D d˛2 a2 D db2 D b3 ˛3 a3 and, consequently, b3 D ˛3 .a3 C da2 /. is shows ˛3 is surjective. We now show ˛3 is injective. Suppose ˛3 a3 D 0. en ˛4 da3 D d˛3 a3 D 0. Since ˛4 is injective, this implies da3 D 0 and hence there exists a2 2 A2 so a3 D da2 since the sequence is exact at A3 . We have d˛2 a2 D ˛3 da2 D ˛3 a3 D 0 : Since the sequence is exact at B2 , there is b1 2 B1 so ˛2 a2 D db1 . Since ˛1 is surjective, we can express b1 D ˛1 a1 . erefore, ˛2 a2 D db1 D d˛1 a1 D ˛2 da1 . Since ˛2 is injective, a2 D da1 . Consequently, a3 D da2 D 0 and ˛3 is injective. We remark that we only used ˛1 is surjective, ˛2 is bijective, ˛4 is bijective, and ˛5 is injective so the hypotheses can be weakened slightly. 8.1.6 RING STRUCTURES IN COHOMOLOGY. Let R D fR0 ; R1 ; R2 ; : : : g be a collection of real (or complex) vector spaces. Denote a generic element of Ri by xi .

1. We say that R is a graded commutative algebra if we have bilinear multiplication maps ? from Rj Rk to Rj Ck which are associative and skew-commutative: xj ? .xk ? x` / D .xj ? xk / ? x`

and xj ? xk D . 1/j k xk ? xj :

2. If R and RQ are graded commutative algebras, we may deﬁne S WD R ˝ RQ by setting Si D ˚pCqDi Rp ˝ RQ q

and .xi ˝ xQj / ? .xk ˝ xQ ` / WD . 1/j k .xi ? xk / ˝ .xQj ? xQ ` / :

118

8. OTHER COHOMOLOGY THEORIES

3. We say a graded commutative algebra R is connected and unital if R0 D R 1

and

for any xj 2 Rj :

1 ? xj D xj ? 1

If R and RQ are connected and unital, then R ˝ RQ is connected and unital where 1R˝RQ WD 1R ˝ 1RQ :

4. If R and RQ are two graded, commutative, connected, and unital algebras, then we say that a collection T D fTi g of linear maps Ti W Ri ! RQ i is an algebra morphism if and T0 .1R / D 1RQ :

Tj Ck .xj ? xk / D Tj .xj / ? Tk .xk /

5. We say that C is a graded commutative cochain complex if C is a graded commutative algebra, and if the diﬀerential d satisﬁes d.xp ? xq / D dxp ? xq C . 1/p xp ? dxq . A morphism in this context is a graded commutative algebra morphism which commutes with the diﬀerential. Lemma 8.5 If C is a graded commutative algebra which is a cochain complex, then the map Œx ? Œy WD Œx ? y gives the cohomology H .C / the structure of a graded commutative algebra. If T W C ! CQ is a morphism of graded commutative algebra cochain complexes, then the map T

in cohomology is a morphism of graded commutative algebras.

Proof. If dxp D 0 and if dyq D 0, then d.xp ? yq / D dxp ? yq C . 1/p xp ? dyq D 0. Consequently, Œxp ? yq is a cohomology class. If xp D dzp 1 , then d.zp

1

? yq / D dzp

1

? yq C . 1/p

1

zp

1

? dyq D xp ? yq C 0 :

erefore, Œxp D 0 in cohomology implies Œxp ? yq D 0 in cohomology. A similar argument shows that if Œyq D 0 in cohomology, then Œxp ? yq D 0 in cohomology. Consequently, the map .Œxp ; Œyq / ! Œxp ? yq

is well-deﬁned and gives a bilinear map from the tensor product H p .C / ˝ H q .C / to H pCq .C /. e identity xp ? yq D . 1/pq yq ? xp implies the multiplication is skew-commutative. A similar argument shows that the multiplication is associative. e assertion about morphisms follows similarly. 8.1.7 THE MAYER–VIETORIS SEQUENCE. We complete the proof of eorem 5.2 by showing the existence of the Mayer–Vietoris sequence in de Rham cohomology and by showing de Rham cohomology is a homotopy functor; this veriﬁes that de Rham cohomology satisﬁes the Eilenberg–Steenrod axioms [17, 18]. We restate these two properties for ease of reference.

8.1. HOMOLOGICAL ALGEBRA

119

eorem 8.6

1. If Oi are open subsets of M with M D O1 [ O2 , then there is a natural long exact sequence (called the Mayer–Vietoris sequence [41, 59]): i1 ˚i2

p 1 ! HdR .M / ! H p

1

.O1 / ˚ H p

1

j1 j2

v

p 1 p .O2 / ! HdR .O1 \ O2 / !HdR .M / !

where we take the natural inclusions: i1 W O1 ! M;

i2 W O2 ! M;

j1 W O1 \ O2 ! O1 ;

j2 W O1 \ O2 ! O2 :

e map v is called the connecting homomorphism; if f W N ! M and if Ui WD f associated open cover of N , then vN f D f vM , i.e., v is natural in this category.

1

Oi is the

p p 2. If fi W M ! N are homotopic smooth maps, then f0 D f1 W HdR .N / ! HdR .M /.

Proof. To prove Assertion 1, we let O1 and O2 be open subsets of M with M D O1 [ O2 . Let D./ denote the de Rham complex d0

d1

0 ! C 1 .0 .// !C 1 .1 .// !C 1 .2 .// ! :

We consider the sequence of cochain complexes: i1 ˚i2

j1 j2

0 ! D.M / ! D.O1 / ˚ D.O2 / ! D.O1 \ O2 / ! 0 :

If we can show this is a short exact sequence of cochain complexes, then Assertion 1 will follow from Lemma 8.3. Let p 2 C 1 .p .M //. If ii p D 0, then p vanishes on Oi . If p vanishes on O1 and on O2 , then p vanishes on M . Consequently, .i1 ˚ i2 /p D 0 if and only if p D 0. is veriﬁes exactness of the sequence at D.M /. We have: .j1

j2 / ı .i1 ˚ i2 / D j1 i1

j2 i2 D .i1 j1 /

.i2 j2 / D 0

since i1 j1 D i2 j2 is just the inclusion of O1 \ O2 in M . Conversely, suppose j1 1 j2 2 D 0. is means the restriction of 1 agrees with the restriction of 2 to O1 \ O2 . Consequently, we may deﬁne to be 1 on O1 and 2 on O2 . is deﬁnes a p -form on M which restricts to i on Oi . is veriﬁes exactness of the sequence at the middle term D.O1 / ˚ D.O2 /. Finally, we must verify that j1 j2 is surjective. We use an argument which we will employ subsequently when considering sheaf cohomology. Let f1 ; 2 g be a partition of unity subordinate to the cover fO1 ; O2 g of M . Let 2 C 1 .p .O1 \ O2 //. Deﬁne: ( ) 2 .x1 /.x1 / if x1 2 O1 \ O2 1 .x1 / WD , 0 if x1 2 O1 \ O2c ( ) 1 .x2 /.x2 / if x2 2 O1 \ O2 2 .x2 / WD . 0 if x2 2 O1c \ O2

120

8. OTHER COHOMOLOGY THEORIES

We must verify i is smooth on Oi . Since 1 D 2 on O1 \ O2 , 1 is smooth on O1 \ O2 . Suppose x 2 O1 \ O2c . e support of 2 is contained in O2 . Let U WD O1 \ supportf2 gc . en x 2 U and 2 D 0 on U . Consequently, 1 is smooth on U as well. Since fO1 \ O2 ; U g forms an open cover of O1 , 1 2 C 1 .p .O1 //; similarly one has that 2 2 C 1 .O2 /. Since j1 1

j2 . 2 / D .2 C 1 / D

the sequence is exact at D.O1 \ O2 /. is establishes Assertion 1. To establish Assertion 2, we shall construct a suitable chain homotopy and apply Lemma 8.1. Let X 2 C 1 .TM / and let 2 C 1 .p M /. Deﬁne int.X/ 2 C 1 .p 1 M / by: fint.X/ g.X2 ; : : : ; Xp

1/

D .X; X1 ; : : : ; Xp

1/ :

(8.1.b)

For example, if xE D .x 1 ; : : : ; x m / is a system of local coordinates, then ( ) 0 if i > 1 1 [email protected] 1 /fdx i1 ^ ^ dx ip g D : dx i2 ^ ^ dx ip if i1 D 1 is is also often called the hook product and is the dual of the interior product int./ for 2 T M deﬁned in Section 5.2. Let F W M Œ0; 1 ! N and let 2 C 1 .p N /. Deﬁne Z 1 .x/ WD [email protected] t /F g.xI t/dt 2 C 1 .p 1 M / : 0

is is invariantly deﬁned. Introduce local coordinates .x 1 ; : : : ; x m I t / on M Œ0; 1. Expand X X Q J .xI t/dt ^ dx J : F D I .xI t /dx I C jI jDp

jJ jDp 1

We show that is the desired chain homotopy by computing: X .f1 f0 / D fI .xI 1/ I .xI 0/gdx I , d D

X

jI jDp

I;i

@x i I dx i ^ dx I C f0 /

D 0 C .f1 d D

XZ i;J

0

X

I dx I

I 1

X I

X

@ t I dt ^ dx I

Z

1 0

J;i

@x i Q J dt ^ dx i ^ dx J

@x i Q J .xI t/dt dx i ^ dx J ,

@x i Q J .xI t /dt dx i ^ dx J .

It is now immediate that d C d D f1

f0 so Assertion 2 follows from Lemma 8.1.

8.2. SIMPLICIAL COHOMOLOGY

8.2

121

SIMPLICIAL COHOMOLOGY

We begin by deﬁning the notion of a ﬁnite simplicial complex K and the associated realization jKj; K is a combinatorial object and jKj is a compact metric space. Although much of what we will say works for inﬁnite simplicial complexes, more care needs to be taken with the topology involved and the arguments that we shall give by induction either fail in the more general setting, or need to be reformulated. 8.2.1 SIMPLICIAL COMPLEXES. A ﬁnite simplicial complex consists of ﬁnite set of vertices V D fv0 ; : : : ; v` g together with a collection K of subsets of V so that the empty set belongs to K , the singleton set fvg for any v 2 V belongs to K , and if A 2 K and if B is any subset of A, then B 2 K . e vertex set V D V .K/ is the union of the singleton sets in K so it need not be speciﬁed separately. Let K be a ﬁnite simplicial complex. Let C0PL .K/ be the ﬁnite-dimensional R vector space with basis V .K/. We introduce a positive deﬁnite inner product on C0PL .K/ by requiring that the vertices form an orthonormal basis. is makes C0PL .K/ into an inner product space and deﬁnes P a topology on C0PL .K/. If x 2 C0PL .K/, expand x D v2V .K/ x.v/ v where x.v/ 2 R are the coeﬃcient functions for v 2 V . Let support.x/ D fv W x.v/ ¤ 0g. Deﬁne the realization of K by setting: X PL jKj WD x 2 C0 .K/ W x.v/ 0 8 v; x.v/ D 1; support.x/ 2 K : v2V .K/

e line segment I.vi1 ; vi2 / with vertices fvi1 ; vi2 g is parameterized by t 1 vi1 C t 2 vi2

for

0 t 1;

0 t 2;

t1 C t2 D 1 :

Note that I.vi1 ; vi2 / jKj if and only if fvi1 ; vi2 g 2 K . e triangle T .vi1 ; vi2 ; vi3 / with vertices at fvi1 ; vi2 ; vi3 g is parameterized by t 1 vi1 C t 2 vi2 C t 3 vi3

for 0 t 1 ;

0 t 2;

0 t 3;

and t 1 C t 2 C t 3 D 1 :

Note that T .vi1 ; vi2 ; vi3 / jKj if and only if fvi1 ; vi2 ; vi3 g 2 K . Consequently, we may think of jKj as a children’s toy where we glue in edges, faces, etc. according to the combinatorial recipe provided by K . We refer to H. Whitney [62, 63] for the proof of the following result as it is a bit beyond the scope of this book. eorem 8.7 Let M be a compact manifold. en there exists a ﬁnite simplicial complex K so that M is homeomorphic to jKj.

Let jV j D m C 1. If K D 2V is the collection of all subsets of V , then jKj is homeomorphic to the unit disk D m in Rm . If m D 1, then jKj is the interval; if m D 2, then jKj is the solid

122

8. OTHER COHOMOLOGY THEORIES

triangle; if m D 3, then jKj is the solid tetrahedron. Let L be the collection of all proper subsets of V . en jLj is homeomorphic to the boundary of D m , i.e., to the sphere S m 1 . We now discuss simplicial cohomology; this is a purely combinatorial object. Let K be a ﬁnite simplicial complex. We deﬁned C0PL .K/ to be the real vector space with basis the vertices of K . If A D fvi0 ; : : : ; viq g is a q -simplex of K , let vA WD vi0 ^ ^ viq 2 qC1 .C0PL .K//,

CqPL .K/ WD spanA2K;jAjDqC1 fvA g qC1 .C0PL .K// .

e fvA g for i0 < < iq form an orthonormal basis for qC1 .C0PL .K// with the induced inner product. Let ıPL WD

X

int.v/;

ıPL .vA / WD

v2V

q X

. 1/j vi0 ^ ^ vij

j D0

1

^ vij C1 ^ ^ viq :

We dualize and set q CPL .K/ WD Hom.CqPL .K/; R/

q 1 q and dPL D ıPL W CPL .K/ ! CPL .K/ :

We have chosen to use the notation ıPL for the boundary operator so that the coboundary operator on the associated cochain complex will be dPL ; this is somewhat diﬀerent notation than is usually employed. Lemma 8.8

2 2 If K is a simplicial complex, then ıPL D 0 and dPL D 0.

2 Proof. By Lemma 5.8, int.v/ int.w/ C int.w/ int.v/ D 0. We will show that ıPL D 0; we then 2 2 2 have dually dPL D .ıPL / D .ıPL / D 0. We compute: X X 2 2 ıPL .A/ D int.v/ int.w/vA D int.w/ int.v/vA D ıPL .vA / . t u v;w2V

2 Because dPL

of K by setting:

D

v;w2V

0, .CPL .K/; dPL / forms a cochain complex and we deﬁne the PL cohomology

q HPL .K/ WD

q qC1 kerfdPL W CPL .K/ ! CPL .K/g

q 1 q rangefdPL W CPL .K/ ! CPL .K/g

:

8.2.2 SIMPLICIAL MAPS. Let K and L be simplicial complexes. A simplicial map is a map of the vertex sets f W VK ! VL so that if A 2 K , then f .A/ 2 L. Let f .vi0 ^ ^ vip / WD f .vi0 / ^ ^ f .vip / :

Since ıL f D f ıK , dK f D f dL . Consequently, we have a map of cochain complexes: .K/; dK / : f W .CPL .L/; dL / ! .CPL

8.2. SIMPLICIAL COHOMOLOGY

123

Lemma 8.1 yields corresponding maps in PL cohomology. We use the inner product on q q .C0PL .K// to identify CPL .K/ with CqPL .K/; under this identiﬁcation dPL is the linear adjoint of ıPL . We deﬁne the PL Laplacian by setting PL WD dPL ıPL C ıPL dPL :

We say K1 is a subsimplicial complex of K if K1 K is a simplicial complex in its own right where the vertex set V .K1 / are just the singletons of K1 , i.e., V .K1 / D K1 \ V .K/. If K1 and K2 are subsimplicial complexes of K , then K1 \ K2 is a subsimplicial complex of K with corresponding P vertex sets V .K1 / \ V .K2 /. Finally, let PL .K/ D ;¤A2K . 1/jAj be number of vertices minus number of edges plus number of triangles etc.; this is the combinatorial Euler characteristic. One has the following result. eorem 8.9

Let K be a ﬁnite simplicial complex.

p 1. If f W K ! L is a simplicial map, then f induces a natural map in cohomology from HPL .L/ p p to HPL .K/ and makes HPL ./ into a contravariant functor.

2. Let K1 and K2 be subsimplicial complexes of K so K D K1 [ K2 . ere is a natural long exact sequence (Mayer–Vietoris) i1 ˚i2

j1 j2

v

p 1 p 1 p 1 p 1 p ! HPL .K/ ! HPL .K1 / ˚ HPL .K2 / ! HPL .K1 \ K2 / !HPL .K/ !

where i1 W K1 ! K , i2 W K2 ! K , j1 W K1 \ K2 ! K1 , and j2 W K1 \ K2 ! K2 are the natural inclusions. e map v is called the connecting homomorphism. 3. If K is the disjoint union of two simplicial complexes K1 and K2 , then q q q HPL .K/ D HPL .K1 / ˚ HPL .K2 / . P q q q 4. HPL .K/ D kerfPL g and PL .K/ D q . 1/q dimfHPL .K/g. ( ) R if q D 0 q S 5. If K D 2 where S is ﬁnite, then HPL .K/ D . 0 if q > 0 Proof. Assertion 1 is immediate from the deﬁnition. We may use Lemma 8.3 to establish Assertion 2 since the short exact sequence of chain complexes 0 ! CPL .K1 \ K2 / ! CPL .K1 / ˚ CPL .K2 / ! CPL .K/ ! 0

gives rise dually to a corresponding short exact sequence of cochain complexes: .K1 \ K2 / ! 0 : .K2 / ! CPL .K1 / ˚ CPL .K/ ! CPL 0 ! CPL

Assertion 3 is again immediate from the deﬁnition as the cochain complexes decouple. e ﬁrst identity of Assertion 4 follows immediately from Lemma 8.2 and the remaining identity follows

124

8. OTHER COHOMOLOGY THEORIES

P i from Lemma 8.2 since PL .K/ D i . 1/i dimfCPL .K/g. We argue as follows to prove Assertion 5. Let A 2 K with jAj 2. en X X ıPL .vA / D int.v/vA and dPL .vA / D ext.v/vA : v2V

v2V Wfa;Ag2K

qC1 e restriction that fa; Ag 2 K is, of course, necessary to ensure that ext.v/vA 2 CPL .K/. But S if K D 2 , the restriction is unnecessary and Lemma 5.8 yields X qPL vA D fext.v/ int.w/ C int.w/ ext.v/gvA D jS jvA : v;w2S

q Consequently, kerfqPL g D f0g and HPL .K/ D 0 if q > 0. We use Assertion 4 to see that: X q 0 PL .K/ D . 1/1 dimfHPL .K/g D dimfHPL .K/g : q

Let jS j D m. We complete the proof of Assertion 5 by nothing that: m PL .K/ D m C m C C . 1/m m D 1. 2 3 m

8.3

SINGULAR COHOMOLOGY

If K and L are simplicial complexes, and if f W K ! L is a simplicial map, there is a natural induced map jf j W jKj ! jLj deﬁned by setting jf j.t i vi / D t i f .vi /. Let Sq D fe0 ; : : : ; eq g, let Kq D 2Sq , and let q WD jSq j be the standard q -simplex. Let j be the face map from Sq 1 to Sq deﬁned by: j .ei / WD ei if i < j and j .ei / D eiC1 if i j : e map j is a simplicial map and jj j W q 1 ! q . In other words, we put the q 1 simplex Sq 1 into the q simplex Sq by laying it along the q 1 simplex obtained by deleting the i th vertex. Consequently, for example, the boundary operator ıPL is given by q X ıPL .Sq / D . 1/i i .Sq

1/ :

iD0

If q D 1, then 0 .0/ D e1 and 1 .0/ D e0 so the boundary of the 1-simplex .e0 ; e1 / is simply .e1 / .e0 /. In terms of Stokes’ eorem, q X bd.Sq / D jıPL j.jSq 1 j/ D . 1/j jj j.jSq iD0

1 j/

where we keep track of the orientation corresponding to the ordering of the vertices. Let X be a topological space. Let CqTP be the R vector space with basis the continuous maps fq W q ! X .

8.3. SINGULAR COHOMOLOGY

e boundary map ıTP from

CqTP .X/

to

CqTP1 .X /

ıTP .fq / D

q X

125

is deﬁned by

. 1/j fq ı jj j :

j D0

2 An easy calculation along the lines of those performed previously shows that ıTP D 0. Let q q TP TP CTP .X/ WD Hom.Cq ; R/ and let h; i be the pairing between CTP .X / and Cq .X /; if ! beq q 1 longs to CTP .X/ and belongs to CqTP .X/, then h!; i 2 R. e derivative dTP from CTP .X/ q q TP to CTP .X/ is deﬁned dually to be dTP ; if ! 2 CTP .X/ and if 2 CqC1 .X /, we have

We have

2 ıTP

D 0 so dually

2 dTP

hd!; i D h!; ı i : D 0. We deﬁne the singular cohomology groups to be:

q HTP .X/ WD

q qC1 kerfdTP W CTP .X / ! CTP .X /g

q 1 q .X / ! CTP rangefdTP W CTP .X /g

:

If f W X ! Y is a continuous map, we set f D f ı and extend linearly to deﬁne a chain map f , which is called pushforward from .CqTP .X /; ıX / to .CqTP .Y /; ıY /; the dual cochain map f q q from .CTP .Y /; dY / to .CTP .X/; dX / is called pullback. eorem 5.2 generalizes to this setting; q HTP ./ satisﬁes the Eilenberg–Steenrod axioms. We shall omit the proof and instead refer to Eilenberg and Steenrod [17, 18] and to Spanier [56]. eorem 8.10

Let X and Y be topological spaces.

p p 1. If f W X ! Y , then f W HTP .Y / ! HTP .X/; Id D Id and .f ı g/ D g ı f . 0 2. If X consists of a single point, then HTP .X/ D R if p D 0 and 0 if p > 0.

3. If Oi are open sets with X D O1 [ O2 , then there is a natural long exact sequence (called the Mayer–Vietoris sequence [41, 59]): i1 ˚i2

j1 j2

v

p 1 p 1 p 1 p 1 p ! HTP .X/ ! HTP .O1 / ˚ HTP .O2 / ! HTP .O1 \ O2 / !HTP .X/ !

where i1 W O1 ! X , i2 W O2 ! X , j1 W O1 \ O2 ! O1 , and j2 W O1 \ O2 ! O2 are the natural inclusions. e map v in the Mayer–Vietoris sequence is called the connecting homomorphism. p p 4. If fi W X ! Y are homotopic maps, then f0 D f1 W HTP .Y / ! HTP .X /.

Let K be a ﬁnite simplicial complex. Order the vertices of K . Let A D fvi0 ; : : : ; viq g be a q -simplex of K where vi0 < < viq . Deﬁne a continuous map A from the standard q -simplex with vertices fe0 ; : : : ; eq g to jKj by setting: A .t0 e0 C C tq eq / WD t0 vi0 C C tq viq :

e map A ! A deﬁnes a linear map K from CqPL .K/ to CqTP .jKj/ which is a chain map. q q .K/ is a cochain map. .jKj/ ! CPL Dually, we have K W CTP

126

8. OTHER COHOMOLOGY THEORIES

eorem 8.11 Let K be a ﬁnite simplicial complex. e cochain map K from CTP .jKj; d / to q q CPL .K; d / deﬁnes a natural isomorphism in cohomology from HTP .jKj/ to HPL .K/.

Proof. Suppose that K has only one vertex v0 so K D f;; v0 g consists of the empty set and a q q singleton set. We then have that HPL .K/ D HTP .jKj/ D 0 for q > 0 so K is trivially an iso0 0 morphism in these degrees. Furthermore, CPL .K/ D CTP .jKj/ D R, K is the identity map, and dPL D dTP D 0. erefore, in this instance, eorem 8.11 follows for this special case. If there are no simplices of higher dimension, then K consists solely of the empty set and singleton sets so K D f;; V g where V D V .K/ is the set of vertices. Let Kv D f;; fvgg for v 2 V . e problem decouples and we have HPL .K/ D ˚v2V HPL .Kv /;

HTP .K/ D ˚v2V HTP .Kv /;

K D ˚v2V Kv

so eorem 8.11 follows from the case K D Kv for a single vertex which was considered above. We therefore may proceed by induction on the cardinality jKj and assume K contains a q -simplex for q > 0. Choose S 2 K so jSj > 0 is maximal. Let A D 2S ;

BDK

S;

C DA

S DA\B:

Let A WD jAj, B WD jBj, C WD jC j, and K WD jKj. We consider the commutative diagram deﬁned by the maps A , B , C , and K : q 1 q 1 q 1 q q q HTP .A/ ˚ HTP .B/ ! HTP .C/ ! H q .K/ ! HTP .A/ ˚ HTP .B/ ! HTP .C/ A ˚ B # ı C # ı K # ı A ˚ B # ı C # q 1 q 1 q 1 q q q q HPL .A/ ˚ HPL .B/ ! HPL .C / ! H .K/ ! HPL .A/ ˚ HPL .B/ ! HPL .C / :

e bottom row is part of a long exact sequence. If we could show that the top row was part of a long exact sequence, we could then use the 5-Lemma (see Lemma 8.4) to show that K was an isomorphism and complete the proof of eorem 8.11. Let A WD fx 2 K W support.x/ \ S ¤ ;g, B WD K fptg where fptg is a point in the interior of A, and C WD A \ B be small open neighborhoods of A, B, and C which deformation retract to A, B, and C. Let iA , iB , iC , and iK be the natural inclusions. We could then consider the diagram: q 1 q 1 q 1 q q q HTP .A/ ˚ HTP .B / ! HTP .C / ! H q .K/ ! HTP .A/ ˚ HTP .B / ! HTP .C / # iA ˚ iB ı # iC ı # iK ı # iA ˚ iB ı # iC q q q q 1 q 1 q 1 .C/ : .B/ ! HTP .A/ ˚ HTP .C/ ! H q .K/ ! HTP .B/ ! HTP .A/ ˚ HTP HTP

e top line is part of a long exact sequence by Mayer–Vietoris. Consequently, the bottom line is part of a long exact sequence. is completes the proof of eorem 8.11. A simplicial map f from K to L preserving the vertex orderings induces maps both in simplicial cohomology and in singular cohomology; one can trace through the isomorphism to

8.3. SINGULAR COHOMOLOGY K fTP

127

fPL K

see that D so we have a natural equivalence of functors if we work in the category of simplicial complexes with an ordering on the vertex set. Let Cq1 .M / be the subspace of CqTP .M / generated by the piecewise smooth maps of the q standard simplex q into M and let C1 .M / be the dual space. e following fact is proved by standard smoothing arguments; we omit the proof in the interests of brevity. Let M be a smooth manifold. e inclusion map of .Cq1 .M /; ı/ into .CqTP .M /; ı/ q and the dual map from .CqTP .M /; d / to .C1 .M /; d / deﬁnes a natural equivalences of functors

Lemma 8.12

q q Hq1 .M / !HqTP .M / and HTP .M / !H1 .M / .

Let M and N be compact smooth manifolds without boundary. Let ! 2 C 1 .p M /. We q deﬁne .!/ 2 C1 .M / by deﬁning .!/ on the generators. If f W q ! M is a piecewise smooth map from the standard q -simplex to M , we deﬁne Z h .!/; f i D .f !/ : q

We apply Stokes’ eorem to see that Z Z h .d!/; f i D .f d!/ D q

q

df ! D

Z bd q

f ! D hf !; bd q i

D h!; ıf i D hdf !; f i : p is shows that is a cochain map from .C 1 .P M /; d / to .C1 .M /; d / and we extend to a map in de Rham cohomology: p p W HdR .M / ! H1 .M / :

If is a smooth map from M to N , then we have a commutative diagram: M

p p HdR .M / ! H1 .M / : "f ı "f N p p HdR .N / ! H1 .M /

Consequently, this is a natural transformation of functors. We can now establish a theorem of de Rham [15]. Recall that a ﬁnite open cover U D fOi gi2A of a manifold M is said to be a simple cover if for any subset B of the indexing set A, the intersection \i2B Oi is either contractible or empty. By eorem 5.7, every compact manifold admits a ﬁnite simple cover. q q .M / in the category is a natural equivalence of functors between HdR .M / and H1 of smooth manifolds without boundary which admit ﬁnite simple covers.

eorem 8.13

128

8. OTHER COHOMOLOGY THEORIES

Proof. Suppose M is contractible. en the homotopy axiom in de Rham cohomology and smooth singular cohomology yields: q q q q HdR .M / D HdR .pt/ D H1 .pt/ D H1 .M / D 0

for

q 1:

0 0 On the other hand, HdR .M / D Œ1 and H1 R.M / D Œ11 where 1 is the constant 0-form and where 11 .pt/ D 1 for any point of M . Since fptg 1 D 1, the desired isomorphism follows. Note that integration is natural with respect to pullback. erefore, we have a commutative diagram of short exact sequences in Mayer–Vietoris and hence the connecting homomorphism in Mayer– Vietoris commutes with integration as well. We apply induction over the number of sets in the ﬁnite simple cover fO1 ; : : : ; O` g. Let U1 WD [i 0g

be the star of a vertex v . Let U WD f?.v1 /; : : : ; ?.v` /g be a ﬁnite open cover of jKj. If I V , the corresponding open set ?.I / WD ?.vi0 / \ \ ?.vip / D fx 2 jKj W ti .x/ > 0 if i 2 I g

is non-empty if and only if I 2 K . is set is contractible and deformation retracts to the barycen1 ter pC1 fvi0 C C vip g of I . We let U D U .K/ WD f?.vi /g; this is a ﬁnite simple cover of M . Let fei g1i` be the standard basis for R` . Let e I WD e i0 ^ ^ e ip . Identify fp in P C p .U ; S q / with the sum jI jDpC1 fp .I /e I : We then have ı.fp / D

` X

X

iD1 jI jDpC1

fp .I /e i ^ e I :

P e relation ı 2 D 0 is then simply the relation that i;j e i ^ e j D 0. Let fi g be a partition of unity subordinate to U . Since i vanishes on Oic , we may extend i f .I / to C 1 .q .?.i0 / \ \ ?.i 1 / \ ?.iC1 / \ \ ?.ip /// to be zero on ?.i /c . is is

8.4. SHEAF COHOMOLOGY

131

exactly the construction we used when establishing the Mayer–Vietoris sequence in eorem 5.2. Let int be interior multiplication, as discussed in Equation (8.1.b). Deﬁne .fp / WD

` X X D1 jI jDp

i f .I / int.e i /e I 2 C p

1

.S q / :

Suppose p > 0 so that even after deleting an index, there is still an index left. We compute ext.e j / int.e i / C int.e i / ext.e j / D ı ij Id; X X fı C ı g D i fext.e j / int.e i / C int.e i / ext.e j /g D i D Id : i;j

i

erefore, provides a chain homotopy that shows H p .S q / D 0 for p > 0; the existence of a “partition of unity” means that S is a ﬂabby sheaf. is proves H p .U ; S q / D 0

if q 0 and if p 1 :

(8.4.b)

Let q > 0. We have the following short exact sequence: i

d

0 ! C 1 .q .O// \ kerfd g !C 1 .q .O// !C 1 .qC1 .OI // \ rangefd g ! 0 : q Because ?.I / is either empty or contractible, HdR .?.I // D 0 for q > 0. is implies that rangefd g D kerfd g so we get a short exact sequence of chain complexes:

0 ! C .U ; Kq / ! C .U ; S q / ! C .U ; KqC1 / ! 0

if

q > 0:

Lemma 8.3 then yields a long exact sequence in cohomology. Equation (8.4.b) implies: 0 D H p .U ; S q / ! H p .U ; KqC1 / ! H pC1 .U ; Kq / ! 0 D H pC1 .U ; S q / for q > 0 and p > 0 :

is proves H pC1 .U ; Kq / D H p .U ; KqC1 /

if q > 0 and if p > 0 :

(8.4.c)

We use Equation (8.4.a) to see that H 0 .U ; S q / D C 1 .q .M //. e beginning of the Mayer–Vietoris sequence then yields that: C 1 .p M / # H 0 .U ; S q /

d

! C 1 .p M / \ kerfd g 0 : ı # # 0 kC1 1 q 1 q ! H .U ; K / ! H .U ; K / ! H .U ; S /

qC1 Consequently, H 1 .U ; Kq / D HdR .M / and a recursive application of Equation (8.4.c) yields: qC1 HdR .M / D H 1 .U ; Kq / D D H qC1 .U ; K0 / :

132

8. OTHER COHOMOLOGY THEORIES

Since each OI is connected, C 1 .0 .O// \ kerfd g consists of the constant functions. erefore, p K0 is the constant sheaf and we have ﬁnally HdR .M / is isomorphic to H p .U I R/. is completes the proof of Assertion 1. It is clear that C p .U ; R/ is a vector space with basis the p -simplices of K . Furthermore, the deﬁnition of the coboundary operator ı in the sheaf-theoretic context agrees with the coboundp ary operator in the PL context. Consequently, H p .U ; R/ D HPL .U /. Assertion 2 now follows; p p Assertion 3, which identiﬁes HPL .U / with HTP .jKj/, was proved in eorem 8.11 using the Mayer–Vietoris sequence.

133

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[59] L. Vietoris, “Über die Homologiegruppen der Vereinigung zweier Komplexe,” Monatshefte für Mathematik 37 (1930), 159–162. 20, 119, 125 [60] F. Warner, “Foundations of diﬀerentiable manifolds and Lie groups,” Scott, Foresman and Co. (1971), Glenview, Ill.-London. 87 [61] R. Weitzenböck, “Invariantentheorie,” Groningen, Noordhoﬀ, 1923. 35 [62] H. Whitney, “Diﬀerentiable manifolds,” Ann. of Math. 37 (1936), 645–680. 121 [63] H. Whitney, “Geometric integration theory,” Princeton University Press (1957), Princeton N.J. 121 [64] W. Ziller, “Lie groups, representation theory and symmetric spaces,” Notes from courses taught Fall 2010 at Penn and 2012 at IMPA, published electronically http://www.math.upenn.edu/ wziller/math650/LieGroupsReps.pdf 87

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Authors’ Biographies PETER B GILKEY Peter B Gilkey¹ is a Professor of Mathematics and a member of the Institute of eoretical Science at the University of Oregon. He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, J. Diﬀerential Geometry and Applications, and J. Geometric Analysis. He received his Ph.D. in 1972 from Harvard University under the direction of L. Nirenberg. His research specialties are Diﬀerential Geometry, Elliptic Partial Diﬀerential Equations, and Algebraic topology. He has published more than 250 research articles and books.

JEONGHYEONG PARK JeongHyeong Park² is a Professor of Mathematics at Sungkyunkan University and is an associate member of the KIAS (Korea). She received her Ph.D. in 1990 from Kanazawa University in Japan under the direction of H. Kitahara. Her research specialties are spectral geometry of Riemannian submersion and geometric structures on manifolds like eta-Einstein manifolds and H-contact manifolds. She organized the geometry section of AMC 2013 (e Asian Mathematical Conference 2013) and the ICM 2014 satellite conference on Geometric analysis. She has published more than 71 research articles and books.

¹Mathematics Department, University of Oregon, Eugene OR 97403 U.S. email: [email protected] ²Mathematics Department, Sungkyunkwan University, Suwon, 440-746, Korea email: [email protected]

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AUTHORS’ BIOGRAPHIES

RAMÓN VÁZQUEZ-LORENZO Ramón Vázquez-Lorenzo³ is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He is a member of the Spanish Research Network on Relativity and Gravitation. He received his Ph.D. in 1997 from the University of Santiago de Compostela under the direction of E. GarcíaRío. His research focuses mainly on Diﬀerential Geometry with special emphasis on the study of the curvature and the algebraic properties of curvature operators in the Lorentzian and in the higher signature settings. He has published more than 50 research articles and books.

³Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain. email: [email protected]

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Index Abelian, 46 Abelian Lie algebra, 47 Ado’s eorem, 54 algebra morphism, 118 almost complex structure, 9 arc-component, 18 Baker, 53 Beltrami, 31 Campbell, 53 Campbell–Baker–Hausdorﬀ formula, 53 Cartan, 54, 114 Cartan–Lie eorem, 54 Cartesian product, 47 Cauchy, 52 Cauchy criteria, 56 Cauchy–Riemann Equations, 9 Chern, 41 Cliﬀord, 26 Cliﬀord algebra, 28 Cliﬀord commutation relations, 28 Cliﬀord multiplication, 27 closed diﬀerential form, 17 closed under bracket, 9 co-associative, 82 co-ring, 82 cochain complex, 112 cochain homotopy, 112 cohomology, 112 combinatorial Euler characteristic, 123

combinatorial Laplacian, 113 compact Kähler manifold, 15 compact Lie group, 68 complex manifold, 10, 12 connected and unital, 118 connecting homomorphism, 114, 115 connection 1-form, 41 contractible, 19 covariant derivative of the curvature operator, 102 Cramer’s rule, 46 de Rham, 17 deformation retract, 19 diagram chasing, 115, 117 Dirichlet, 32 Eilenberg, 114 Einstein convention, 6 Ekaterina Puﬃni, xiii extension, 128 exterior multiplication, 27 ﬁber bundle, 87 ﬁrst Chern class, 42 ﬂabby sheaf, 131 Fourier, 32 G-module, 62 geodesic coordinates, 6 geodesic involution, 104 geodesic spray, 103

142

INDEX

geodesic symmetry, 104 Grassmann, 91 Grassmannian, 90 Hausdorﬀ, 53 Hermitian metric, 13 Hodge, 31, 34 Hodge ? operator, 28 Hodge Decomposition eorem, 34 Hodge–Beltrami Laplacian, 31 holomorphic, 9 holomorphic coordinate system, 9 homogeneous, 98 homogeneous coordinates, 12 homotopic maps, 19 homotopy equivalent spaces, 19 hook product, 120 Hopf, 82 Hopf algebra, 82 Hopf ﬁbration, 90 hyperbolic boost, 74 ideal, 47 inner product, 65 integrable complex structure, 10 interior multiplication, 27 irreducible, 62 isometry, 91 isothermal, 10 Jacobi, 47 Jacobi identity, 47, 96 Jacobi operator, 103 Jacobi vector ﬁeld, 102, 103 Kähler, 13 Kähler form, 13 Kähler manifold, 13, 15 Kähler metric, 15 Künneth, 39

Künneth formula, 39 Killing, 71, 95 Killing form, 71 Killing vector ﬁeld, 96 Krill Institute of Technology, xiii Laplace, 31 Laplacian, 31 left-invariant vector ﬁelds, 50 Leibnitz formula, 95 lens space, 105 Levi–Civita, 5 Lie, 45, 54, 95 Lie algebra, 47 Lie algebra structure constants, 49, 71, 81 Lie bracket, 47, 81 Lie derivative, 95 Lie group, 46 Lie subalgebra, 47, 48 Lie subgroup, 59 linear fractional transformations, 94 Lipschitz, 26, 52 local Killing vector ﬁeld, 99 locally ﬂat, 99 locally homogeneous, 98 long exact sequence, 19, 113 matrix groups, 55 Maurer–Cartan form, 84 Mayer, 20 Mayer–Vietoris sequence, 20 morphism, 111 natural, 115 natural transformation of functors, 54, 111 Newlander–Nirenberg, 9 Nijenhuis tensor, 9, 10 Nirenberg, 10 oriented volume form, 28 orthogonal group, 61

INDEX

perfect pairing, 37 PL cohomology, 122 Plücker, 91 Poincaré, 37 principal bundle, 87 projective special linear group, 94 Puﬃni, Ekaterina, xiii pullback, 17, 125 pullback bundle, 41 pullout diagram, 41 pushforward, 50, 125 quaternions, 46

simple cover, 25, 127 simplicial complex, 121 simplicial map, 122 singular cohomology, 125 skew-commutative, 117 skew-ﬁeld, 46 Snake Lemma, 114 special linear group, 61 star of a vertex, 130 submodule, 62 subsimplicial complex, 123 symmetric space, 104

real projective space, 22 realization, 121 representation, 53, 62 Riemann, 32

tautological line bundle, 12, 42, 43 transvection, 105

sheaf, 128 short exact sequence, 113 short exact sequence of cochain complexes, 114

Vietoris, 20

unitary group, 61

wedge product, 17 Weyl, 68

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