Differentiable Manifolds: Forms, Currents, Harmonic Forms [Softcover reprint of the original 1st ed. 1984] 3642617549, 9783642617546

Table of contents :
Préface à l'édition anglaise
Introduction to the English Edition
Table of Contents
Table of Contents
Introduction
Chapter I. Notions About Manifolds
§1. The Notion of a Manifold and a Differentiable Structure
§2. Partition of Unity. Functions on Product Spaces
Chapter II. Differential Forms
§4. Differential Forms of Even Type
§5. Differential Forms of Odd Type. Orientation of Manifolds and Maps
§6. Chains. Stokes' Formula
§7. Double Forms
Chapter III. Currents
§8. Definition of Currents
§9. The Vector Spaces E, D, E^P and d^{p4}
§10. The Vector Spaces D', E', D^{'P} and E^{'p7}
§11. Boundary of a Current. Image of a Current by a Map
§12. Double Currents
§13. Transformations of Double Forms and Currents by a Map
§14. Homotopy Formulas
§15. Regularization
§16. Operators Associated with a Double Current
§17. Reflexivity of E and D. Regular Operators and Regularizing Operators
Chapter IV. Homologies
§18. Homology Groups
§19. Homologies in R^n
§20. The Kronecker Index
§21. Homologies Between Forms and Chains in a Manifold Endowed with a Polyhedral Subdivision
§22. Duality in a Manifold Endowed with a Polyhedral Subdivision
§23. Duality in Any Differentiable Manifold
Chapter V. Harmonic Forms
§24. Riemannian Spaces. Adjoint Form
§25. The Metric Transpose of an Operator. The Operators δ and ⊿
§26. Expressions of the Operators d, δ and ⊿ Using Covariant Derivatives
§27. Properties of the Geodesic Distance
§28. The Parametrix
§29. The Regularity of Harmonic Currents
§30. The Local Study of the Equation ⊿μ = β. Elementary Kernels
§31. The Equation ⊿S=T on a Compact Space. The Operators H and G
§32. The Decomposition Formula in a Non-Compact Space
§33. Explicit Formula for the Kronecker Index
§34. The Analyticity of Harmonic Forms
§35. Square Summable Harmonic Forms on a Complete Riemannian Space
Bibliography
Subject Index
List of Notation

Citation preview

Grundlehren der mathematischen Wissenschaften 266 A Series of Comprehensive Studies in Mathematics

Editors M. Artin S. S. Chern 1. M. Frohlich E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. Mac Lane W Magnus C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits B. L. van der Waerden M. Waldschmidt S. Watanabe Managing Editors M. Berger B. Eckmann S. R S. Varadhan

Georges de Rham

Differentiable Manifolds Forms, Currents, Harmonic Forms

Translated from the French by F. R. Smith Introduction to the English Edition by S.S.Chern

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Georges de Rham 7, avenue Bergieres CH -1004 Lausanne

Title of the French original edition: Varietes differentiables Publisher Hermann, Paris 1955

AMS Subject Classification (1980): 58Axx, 55Nxx, 57Rxx, 58Gxx ISBN-13: 978-3-642-61754-6 DOl:1O.1007/978-3-642-61752-2

e-ISBN-13: 978-3-642-61752-2

Library of Congress Cataloging in Publication Data Rham, Georges de. Differentiable manifolds. (Grundlehren der mathematischen Wissenschaften; 266) Translation of: Varietes diflerentiables. Bibliography: p. Includes index. \. Differentiable manifolds. 2. Differential forms. 3. Riemannian manifolds. I. Title. II. Series. QA614.3.R4713 1984 514'.3 84-10597 ISBN-13: 978-3-642-61754-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Softcover reprint of the hardcover 1st edition 1984 2141/3140-543210

Preface

a l'edition anglaise

Je tiens a remercier tres sincerement tous ceux qui ont rendu possible cette traduction de mon livre: d'abord Beno Eckmann, c'est grace a lui que Ie projet a ete realise; Shiing-Shen Chern qui a eu la grande amabilite d'ecrire une Introduction; Springer-Verlag pour tous ses efforts et sa cooperation; et Ie traducteur pour son travail consciencieux. Voici quelques complements historiques de nature personelle. Attire vers les Math6matiques des 1924, c'est en 1928 et 1929 que j'ai fait rna these. En 1930, H. Lebesgue, a qui j'apportais mon manuscrit, m'a dirige vers Elie Cartan, qui a bien voulu l'examiner et faire un rapport favorable. Ma these a paru en juin 1931, precedee de deux Notes aux Comptes Rendus en 1928 et 1929. Peu apres, S. Lefschetz ecrit a H. Lebesgue" ... vous no us rendriez grand service en suggerant a M. de Rham de nous envoyer quelques exemplaires de sa these. M. Hodge nous en a expose la partie analytique ... ". Ce fut, semble-t-il, l'origine des travaux de Hodge publies vers 1934-1936. L'enonce de son th60reme frappe par sa beaute et sa simplicite. Mais ses demonstrations m'ont paru tres penibles et trop difficiles. Ce qui m'a amene a reprendre Ie probleme dans les travaux pUblies en 1946. Pendant la guerre, les "Annals of Mathematics" ne parvenaient pas dans nos bibliotheques et j'ai ignore Ie travail de H. Weyl auquel S. S. Chern fait heureusement allusion dans son Introduction. Je n'en ai eu connaissance que plus tard. En 1950, a l'Institute for Advanced Study de Princeton, a la demande precisement de H. Weyl a qui je dois beaucoup pour l'interet qu'il m'a temoigne, j'ai fait une serie d'exposes "Harmonic Integrals". De la, grace a l'interet et l'amitie d'Andre Weil, est issu mon livre. Lausanne, en mai 1984

Georges de Rham

Introduction to the English Edition

William Hodge's theory of harmonic integrals was both bold and imaginative. In one step he found the key to the n-dimensional generalization of geometric function theory. His proof of the fundamental theorem contained a serious gap. This was filled in a masterful way by Hermann Weyl, using his earlier results on potential theory. Professor de Rham's book is an introduction to differentiable manifolds. Its main objective seems to be the first detailed proof, different from Hodge-Weyl, of Hodge's fundamental theorem. It must have given him great pleasure in writing the book, for Hodge theory is a natural culmination of the de Rham theory. In n-dimensional geometry a fundamental notion is the "duality" between chains and cochains, or domains of integration and the integrands. While the boundary operator is a global operator, the coboundary operator, i.e. exterior differentiation, is local. This makes cohomology theory a more convenient tool for analytical treatment and for applications. Poincare recognized the importance of the multiple integrals and stated the main "theorems", while Elie Cartan developed the foundations of the exterior differential algebra and applied it to mechanics, differential systems, and differential geometry. The global theory was completed by de Rham's famous thesis in 1931. The thesis was long, because at that time topology was homology theory and the notion of cohomology did not exist. A notion which includes both chains and co chains is that of a "current". This was introduced by de Rham and used effectively throughout the book. A zero-dimensional current is a distribution (in the sense of Laurent Schwartz), now a fundamental concept in mathematics. There are now other proofs of Hodge's theorem. Perhaps the most natural approach is through pseudo-differential operators; cf. [5], [6]. The MilgramRosenbloom proof using the heat equation method is an idea with broad repercussions [3] Morrey, Eells, and Friedrichs gave a proof using a variational method [4]. Hodge's theorem admits various extensions. The most important one is to cohomology theory with a coefficient sheaf, which was introduced by J. Leray and developed for the complex structure with great success by Henri Cartan and J-P. Serre [1], [6]. Its harmonic theory was first worked out by K. Kodaira [2]. When geometrical information is available, the harmonic theory allows the proof of "vanishing theorems" on cohomology groups, using an idea originated from S. Bochner. Such vanishing theorems are of great importance.

VIII

Introduction to the English Edition

Modern developments in the general area of "elliptic operators on manifolds", such as the index theory and the spectral theory, have raced way beyond the content of this book. I believe, however, that in his enthusiasm for new results a mathematician will be well-advised to stop at this landmark, where he will have a lot to learn both on the mathematics and on the mathematical style. Berkeley, February 1984

S. S. Chern

References [I] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. John Wiley 1978 [2] Kodaira, K.: On a differential-geometric method in the theory of analytic stacks. Proc. Nat. Acad. Sci., USA, vol. 29 (1953), 1268-1273 [3] Milgram, A.N., Rosenbloom, P.C.: Harmonic forms and heat conduction, I, II. Proc. Nat. Acad. Sci., USA, vol. 37 (1951), 180-184,435-438 [4] Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Grundlehren der math. Wiss. 130, Springer 1966 [5] Nirenberg, L.: Pseudo-differential operators, Global Analysis. Proc. Symp. Pure Math., vol. 16, Amer. Math. Soc., 149-167, 1970 [6] Wells, R.O.: Differential Analysis on Complex Manifolds. Prentice Hall, Inc. 1973; second edition, Graduate Texts in Mathematics, no. 65, Springer 1980

Table of Contents

Introduction . . . . . . . . . . . . . . . .

1

Chapter I. Notions About Manifolds.

3

§ 1. The Notion of a Manifold and a Differentiable Structure §2. Partition of Unity. Functions on Product Spaces. §3. Maps and Imbeddings of Manifolds. . . . . . . . . . . . . . . .

3 4 9

Chapter II. Differential Forms §4. Differential Forms of Even Type. . . . . . . . . . . . . . . . . . .. §5. Differential Forms of Odd Type. Orientation of Manifolds and Maps. . . . . . . . . . . . §6. Chains. Stokes' Formula §7. Double Forms . . .

15 19 23 30

Chapter III. Currents . . . . .

34

§8. §9. § 10. § 11. §12. §13. §14. § 15. § 16. § 17.

34 37

Definition of Currents . . . . . . . . . The Vector Spaces g, !?2, gP, and !?2 P The Vector Spaces !?2', g', !?2'P, and g'P Boundary of a Current. Image of a Current by a Map. . . . Double Currents . . . . . . . . . . . . . . . . . . . . . . . . . . Transformations of Double Forms and Currents by a Map. Homotopy Formulas . . . . . . . . . . . . . . . . Regularization . . . . . . . . . . . . . . . . . . . Operators Associated with a Double Current . Reflexitivity of {f and !?2. Regular Operators and Regularizing Operators . . . . . . . . . . . . . . . . . . . . . .

40 45

48 50 56

61 70

73

Chapter IV. Homologies.

79

§ 18. Homology Groups .. § 19. Homologies in JR." ..

79

§20. The Kronecker Index .. §21. Homologies Between Forms and Chains in a Manifold Endowed with a Polyhedral Subdivision. . . . . . . . . . . . . . . . . . . . §22. Duality in a Manifold Endowed with a Polyhedral Subdivision §23. Duality in Any Differentiable Manifold . . . . . . . . . . . . . . .

83

82 87 89 96

X

Table of Contents

Chapter V. Harmonic Forms . . . . . . . . . . . . . . . . . . §24. Riemannian Spaces. Adjoint Form . . . . . . . . . . . . §25. The Metric Transpose of an Operator. The Operators D and Ll §26. Expressions of the Operators d, D, and Ll Using Covariant Derivatives . . . . . . . . . . . . . . . §27. Properties of the Geodesic Distance. . §28. The Parametrix. . . . . . . . . . . . . . §29. The Regularity of Harmonic Currents §30. The Local Study of the Equation Llf.1 = {J. Elementary Kernels §31. The Equation LlS = T on a Compact Space. The Operators Hand G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . §32. The Decomposition Formula in a Non-Compact Space. §33. Explicit Formula for the Kronecker Index. . . . . . . . . §34. The Analyticity of Harmonic Forms . . . . . . . . . . . . §35. Square Summable Harmonic Forms on a Complete Riemannian Space. .

99 99 103 106 111 121 127 129 130 136 144 151 158

Bibliography.

161

Subject Index.

165

List of Notation.

167

Introduction

In this work, I have attempted to give a coherent exposition of the theory of differential forms on a manifold and harmonic forms on a Riemannian space. The concept of a current, a notion so general that it includes as special cases both differential forms and chains, is the key to understanding how the homology properties of a manifold are immediately evident in the study of differential forms and of chains. The notion of distribution, introduced by L. Schwartz, motivated the precise definition adopted here. In our terminology, distributions are currents of degree zero, and a current can be considered as a differential form for which the coefficients are distributions. The works of L. Schwartz, in particular his beautiful book on the Theory of Distributions, have been a very great asset in the elaboration of this work. The reader however will not need to be familiar with these. Leaving aside the applications of the theory, I have restricted myself to considering theorems which to me seem essential and I have tried to present simple and complete proofs of these, accessible to each reader having a minimum of mathematical background. Outside of topics contained in all degree programs, the knowledge of the most elementary notions of general topology and tensor calculus and also, for the final chapter, that of the Fredholm theorem, would in principle be adequate. After the definition of differentiable manifolds, Chapter I establishes some results necessary for the sequel. In particular, the existence of partitions of unity and the theorem of Whitney concerning the problem of embedding a manifold in a Euclidean space are given. Chapter II describes the elements of the theory of differential forms and differential chains together with the exterior differential calculus of E. Cartan and the generalised Stokes' formula. I have introduced and systematically used the notions offorms and chains of even type and odd type. These concepts allow the theory to be applied as well to non-orientable manifolds. Finally, the notion of double form prepares the way for the introduction of some other generalisations. Chapter III is devoted to the definition and to the study of the general properties of currents. Certain properties of topological vector spaces which we require here are recalled along the way together with their proofs. I have introduced double currents, which generalise the kernel distributions of L. Schwartz, and the regularising operators which show in a precise way that a current can always be considered as the limit of a sequence of forms. In Chapter IV, currents are used to define and study the homology groups of a manifold. We will find there complete proofs of theorems which provide a relationship between differential forms and chains from the homological

2

Introduction

viewpoint in a way analogous to the Poincare duality theorem in a differentiable manifold. Chapter V deals with the principles of the theory of harmonic differential forms in a Riemann space. First, the Hodge theorem is established using the integral equations method which in the case of compact spaces leads to the most complete results. The lemmas upon which this method rests are proved in detail. So too are all the properties of the geodesic distance which occur there. The method of orthogonal projection in a Hilbert space which allows us to proceed further in the non-compact case is considered next, and the decomposition theorem of Kodaira is generalised by the introduction of currents. We deduce from these formulas providing us with an integral representation of the Kronecker index of two chains. Next, by the method ofE. E. Levi, we prove that the harmonic differential forms are analytic in an analytic Riemannian space. In particular, it follows that such a form cannot have a zero of infinite order unless it is identically zero. We make note that by virtue ofa theorem ofN. Aronsjan, A. Krzywicki and J. Szarski, this is moreover valid for harmonic forms in a Coo Riemannian space. Finally we conclude with an interesting theorem of A. Andreotti and E. Vesentini which concerns square summable harmonic forms on complete, non-compact Riemannian spaces.

Chapter I. Notions About Manifolds

§ 1. The Notion of a Manifold and a Differentiable Structure An n dimensional manifold is a separable topological space, each point of which has a neighbourhood homeomorphic to an open n dimensional ball. Moreover we shall always suppose that this space admits a countable base of open sets, that is, there exist a countable sequence of open sets such that any open set may be expressed as a union of sets of the sequence. On an n dimensional manifold Va differentiable structure ofinfinite order, or, more briefly, a COO structure, is defined by prescribing at each point x of Va class of real valued functions which are said to be Coo at x such that the following axiom is satisfied. Axiom for Coo Structures. For each point u of V there exists an open neighbourhood U ofu and n real valuedfunctions defined on U, Xl (x), . .. , xn(x), such that: (a) The map X-+(Xl (x), . .. ,xn(x» ofUinto JRn is a homeomorphism ofU onto a connected open subset ofJRn such that each function f defined on U or on a subset of U can be expressed with the aid of Xl> ... ,Xn, f(x) = f(Xl (X), . .. ,Xn(X»; (b) The function f(x) is Coo at a point of U if and only if there exists an open neighbourhood W of this point, contained in U, such thatf(x) is defined on Wand such that f(Xl , ... ,xn) has continuous derivatives of all orders for values of the variables Xl, ... ,Xn corresponding to points of W.

A function is said to be Coo if it is Coo at each point of Vand Coo on a subset A of Vif it is Coo at each point of A. It follows from (b) that the set of all points where a function is Coo is an open set. Every system of n functions Xl, ... , Xn defined in an open set U and having the properties (a) and (b) is called a system of local coordinates in U. Our axiom ensures the existence of such a system in an open neighbourhood of each point. From (b), these coordinates are Coo functions in U. If fl , ... ,In is another system of local coordinates in U, the functions fl, . .. ,In are Coo functions for which the Jacobian with respect to Xl, ... ,Xn is non-zero in U. On the other hand, by virtue of the classical implicit function theorem, if fl, . .. ,In are Coo functions on U for which the Jacobian with respect to Xl, ... ,Xn is =1= 0 at a point u' of U, there exists a neighbourhood U' of u' in which Ji, . .. ,In are local coordinates.

Chapter 1. Notions About Manifolds

4

In practice, a Coo structure is most often defined by an open covering {Ui } of V with a system oflocal coordinates in each Ui' On the intersection Uin Uj, the

coordinates of one system must be infinitely differentiable functions of the coordinates of the other, in accordance with (b). In the space lRn , there exists a canonical Coo structure, given by the system of coordinates Xl , . . . , Xn defining lR n; the Coo functions are infinitely differentiable functions of Xl,· .. , X n . A function defined on a manifold with a Coo structure is said to be cr, r being an integer ~ 0, if, when expressed in terms of the local coordinates, it has continuous derivatives up to and induding order r. A CO function is simply a continuous function. A Coo manifold is a manifold with a Coo structure. As only Coo manifolds are to be considered in the following, we shall call them simply manifoldsl . We may define analogously the notions of differentiable structure of order r and of real analytic structure: it is sufficient to modify condition (b) of the above axiom by demanding only that f(x i , . . . ,xn ) has continuous derivatives up to order r or that it is real analytic. Thus, we obtain the notions of C r manifold and of real analytic manifold.

§ 2. Partition of Unity. Functions on Product Spaces2 Letf(x) be a function equal to exp (_x- 2 ) for x>O and to 0 for x:;:;O, and let x

S g(x) =

f(t)f(1 -t)dt

--00-00------

S f(t)f(1 - t)dt -00

The function h(x)=g(x+1)-g(x) is Coo, (-1,1), and we have

I j= -

~O

and zero outside of the interval

h(x-J) = 1, 00

as is easily verified. By successively replacing X with Xl 8

, . . . , Xn 8

and multiplying

together the relations obtained, it follows that

. I. h (~l J1,···,1n

-JI)

h

(~2 -.iz) ... h (~n -in) = 1.

1 This definition of differentiable manifold is equivalent to that of Whitney, and, in this form, very near to that given by Chevalley (Whitney [1], Chevalley [1], Chapter III). 2 For this §, c.r., Dieudonne [1] and Schwartz [2], p. 23.

5

§2. Partition of Unity. Functions on Product Spaces

Denoting the system of n integer indices jl, . .. ,jn by the positive integers = j(jl, . .. ,jn), the point (Xl, ... ,xn) of lRn by X and putting

j

this relation may be written as (1)

The function cPj(x) is Coo, ~ 0 and vanishes outside the cube of side length 2e and centre at the point (jle, ... ,jne), i.e., outside of the cube defined by the relations IXi-jiel~e (i=1, ... ,n). The support of a continuous function is defined to be the smallest closed set outside of which the function vanishes. The support of cPix) is the cube described above. Let A be a compact set in lRnand B an open set containing A. If e is sufficiently small, the length of the diagonal of the cube of side length 2e is less than the distance of A to the frontier of B, and therefore, if such a cube meets A, it is contained in B. Let us consider then the functions cPix) with support meeting A. There is a finite number of them and their sum is a Coo function with compact support in B, its values lying entirely between 0 and 1 (the endpoints are not excluded) and equal to 1 on A. Thus, we have proved the following proposition.

Lemma. If A is a compact set in lRn and B is an open set containing A, there exists a Coo function with compact support in B, with values lying between 0 and 1 and equal to 1 on A. This lemma will be used to prove the following theorem.

Theorem 1. For any open covering {UJ of a manifold V, where i runs through any set of indices, it is possible to find a collection of functions cPj, where j runs through a finite or countably infinite set of indices, satisfying the following conditions: 1) cPj~O, cPj= 1;

L j

2) cPj is Coo and has compact support contained in one of the Ui ; 3) Each point of V has a neighbourhood which meets only a finite number of the supports of the cPj.

If condition 1) is satisfied, the expression L cPj is called a partition of unity. j

We express condition 2) by saying that this partition is subordinate to the covering Ui and condition 3) by saying that it is locally finite. By virtue of the Borel-Lebesgue theorem, this condition 3) is equivalent to the following: 3') Every compact set meets the supports of only a finite number of the cPj.

Chapter 1. Notions About Manifolds

6

The collection of the CPj will be finite depending on whether V is compact or not. In order to prove this theorem, let us first show that there exists an open covering {G i } of V having the following properties: (a) It is subordinate to the covering {Ui}, that is, the closure Gi of each Gi is contained in one of the Uj • (b) It is locally finite, that is, a compact set can meet only a finite number of the G i ; (c) The closure Gi of each Gi is contained in the domain of a coordinate system. Since it is supposed that the manifold has a countable open basis, we know that from every open covering, a finite or countable covering is able to be extracted. For each point of V, we can find an open set U' which contains it and which has compact closure contained in one of the Ui and in the domain of a coordinate system; consequently, there exists a finite or countable covering {Ui} formed by such sets and this covering possesses properties (a) and (c). Ifit is not locally finite, such a one can be constructed by shrinking the U; in the following way. Consider the sequence of compact sets K 1 , K 2 , . . . and the sequence of integers .i1 , .il, ... defined recurrently by putting

.im=the smallest integer >.im-1 such that Km- 1 C

(J

(J

Uj, Km= Uj. Km- 1 is j=l j=l contained in the interior of Km and any compact set is contained in Km whenever Tn is sufficiently large. We then define the G by putting Gi=U; if i 0 for r > 0 and tending to zero with r, such that, if we denote the linear map tangent to fl at x by M x, IflY - M xYI ~ IY - xlb (I Y - xl)· If xEE, Mx has zero determinant and, when we vary Y only, MxY remains in an (n -1) dimensional plane II. If, in addition IY - xl ~ e, we have IflX - flYI ~ ae, and the distance of flY to II remains ~ eb(e). Thus, the point flY stays in the 4

Sard [1].

§3. Maps and Imbeddings of Manifolds

11

volume cut out in the ball of radius ae and centre J1X by the two planes parallel to the diametral plane n and a distance eb(e) from it. This volume is ~Ab(e)en, where A is a numerical constant (the product of a by the volume of the ball of radius 1 in JRn-1). Partition C into hn equal cubes, of side-length h- 1 and of diameter d = Vnh -1. Each of these cubes containing a point of E is contained in a ball of radius d centred at this point and so, has an image contained in a volume ~Ab(d)dn. The image of all these cubes and, consequently, that of E, is thus contained in a total volume ~Ab(d)nnI2. This volume may be made arbitrarily small if the integer h is sufficiently large. Thus, f.1.E is of measure zero. Denote by R ro the infinite dimensional space formed by the points x = (Xl, X2, . .. ) for which all the coordinates Xi except for finitely many are zero. A map f.1.Y = X, yE V, XE JR ro, of V into JR ro will be said to be Coo if each coordinate Xi of f.1.Y is a Coo function of y in V. A Coo map of the n dimensional manifold V into JRro (or into JRm), f.1.y=x, is called a locally regular imbedding of V into JR ro (or into JRm), if, for each yE V, there exists n indices i1 , • •• ,in such that the n functions Xi(Y) (i = i1 , • •• , in) form a system oflocal coordinates in a neighbourhood U C Vofy. As we have seen, the condition for this to be the case is that the Jacobian of the functions Xi(Y) (i= i1 , • .• ,in) with respect to the local coordinates ofy be non-zero. For f.1. to be a locally regular imbedding, it is then necessary and sufficient that the matrix formed by the first derivatives of the coordinates of X= f.1.y with respect to the local coordinates of y be everywhere of maximum rank n. We say that f.1.is a regular imbedding ifit is a locally regular imbedding and an injection, that is, f.1.y"* f.1.Z whenever y"* z. A map of one manifold into another, in particular, an imbedding, is said to be proper if thepreimage of each compact set is compact. We sometimes express this condition by saying that X =f.1.y-+ 00 if y-+oo.

Theorem 5 (H. Whitney)s. For each n dimensional manifold V, there is a proper and regular imbedding of V into JR2n + 1. First we will establish the existence of a regular imbedding of V into JR ro; this will be the aim of Lemma 1. Next, using projections that a second lemma and Theorem 3 will assure us exist, we will deduce a regular imbedding of V into JR2n+1, and by making use of this, we will finally obtain a proper and regular imbedding of V into JR2n + 1 . Lemma 1. There exists a regular imbedding X = fy of V into JR ro such that the supports ofthefunctions Xi=Xi(Y), the coordinatefunctions off, are compact and such that each compact set K C V meets only a finite number of them. 5 Whitney [1]. Actually, Whitney proves a stronger theorem. He supposes only that the given manifold is C 1 and obtains an imbedding affording an analytic submanifold of IR2. + 1. However, the proof is longer. I know of another proof of the same theorem, due to N. Bourbaki, which uses instead of the notion of sets of measure zero that of meager sets (sets of the first Baire category).

12

Chapter I. Notions About Manifolds

Take the open covering {G i } of V considered in the proof of Theorem 1. This covering is locally finite and each Gi is contained in the domain of a local coordinate system. We can find another open covering { Ui} such that Ui C Gi and Coo functions, Qi,O(Y), such that O~Qi,O(Y) ~ 1 for each yE V, Qi,O(Y) = 1 if yE Ui and the support of Qi,O(Y) is contained in Gi . Let Y1, . .. , Yn be local coordinates in Gi and put

Let us establish an injection between the set of positive integers h and the set of pairs of integers (i,j) such that i ~ 1, 0 ~j ~ n, by putting h = (n + 1)(i -1) +j + 1. Furthermore, put Thus, we define amapJ,fy = (Xl> X2, . .• ), of Vinto lR"'which possesses all the required properties. In fact, the Qi,j(Y) U= 1, ... ,n) form a system oflocal coordinates in Ui and each point yE V is in one of the Ui ; if yE Ui and if fz = fy, we have Qi,O(Z) = Qi,O(Y) = 1, and so, zEG;, and since z has the same local coordinates Qi,iz) = Yh= 1, ... , n) as yin G;, y=z. This shows thatfis a regular imbedding. Finally, a compact set K C V, meeting only a finite number of G;, can meet the D supports of only a finite number of Qi,j(Y). The set of directions oflines * oflRm forms as we know an (m -1) dimensional manifold ]p"'-1 which is usually called the plane at infinity of IRm, and we call point at infinity of a line the point ofIPm - 1 representing its direction. Further, we will denote by IP'" the set of directions of lines in IR"'. As IRmC IR"', we have IPm - 1 C IP"'.

Lemma 2. Letfbea regular imbedding of V into IR'" and let 1R2n +2 bea(2n+2) dimensional subspace of IR'" and JP2n + 1 its plane at infinity. The set of points of IP2n + 1 through which no tangent nor any chord ofjV passes is everywhere dense in IP2n+ 1. The set of points (x,y) of Vx V such that the orthogonal projection onto IR2n +2 of the vector joiningfx to fy is zero is closed. Its complement D forms a 2n dimensional manifold. The map 4> of D into IP2n+\ associating with (x,y) the point at infinity of the orthogonal projection onto 1R2n +2 of the line joiningfx to fy, is Coo, and 4>D contains the set of points ofIP2n+1 through which a chord ofjV passes. This latter set thus has measure zero since, by Theorem 3, 4>D has measure zero.

* Translator's Notes:

1. A "line", in this context, is a straight line of infinite extent. 2. The "set of directions of lines" should be read as the set of equivalence classes of lines under the relation of parallelism. This is the classical definition of the "ideal" or projective space associated with a vector space.

13

§3. Maps and Imbeddings of Manifolds

Now, let V' be the space of directions of tangent vectors to V, a fibration with base V. It is a (2n -1) dimensional manifold. Since f is a locally regular imbedding, each non-zero tangent vector to V has a non-zero image under f in lR W andfextends to a mapl' of V' into ]pw. The set of points of V' having as image underl' a direction orthogonal to lR2n +2 is closed and its complement D' forms a (2n -1) dimensional manifold. The map !/J of D' into ]p2n +1, associating with each Y E V' the direction of the orthogonal projection onto lR2n +2 of any line passing through l' Y, is a Coo map and !/J D' contains the set of points of ]p2n +1 through which a tangent offV passes. As above, we deduce from this that this set has measure zero in ]p2n +1. Since the union of two sets of measure zero has measure zero, the set of points of ]p2n + 1 through which a tangent or a chord of fV passes is of measure zero and its complement is everywhere dense. This proves Lemma 2. D Now, we note that there exist proper Coo maps of Vinto lR 2n +1, and even into lR l . In fact, if 1 =

00

L CPi is a partition of unity in V satisfying the conditions of

i=l

00

Theorem 1, we define such a map by putting

Xl

=

I

iCPi. Theorem 5 is then a

i=l

consequence of the following proposition which we will prove: Iffis aproper Coo map ofVinto lR2n + 1 , then,for any 8 > 0, there exists a regular imbedding g of V into lR 2n + 1 such thatJor all YEV, the distance offy to gy is < 8.

The imbedding g will necessarily be proper whenever f is. Let tRw be the subspace of lR OJ defined by Xi = 0 for 1 ~ i ~ 2n + 1, so that lR OJ =lR2n +1 X tRw, and letl' be a regular imbedding of V into tRw satisfying the conditions of Lemma 1, and moreover such that the distance offy to the origin is bounded. This can always be achieved by dividing each coordinate ofl'Y by the product of the least upper bound of its modulus by its sign. The mapfo = (f,f) of Vinto lR W is then a regular imbedding and the orthogonal projection offoY into lR 2n + 1 isfy· Denote by lR h the subspace of lRW defined by Xi = 0 for i> h, by lRk' the subspace of lR W defined by Xi = 0 for 2n + 1 < i~2n + 1 + k, and by ]ph-1 and ]Pk' their planes at infinity. Let poE]p2n+1 be a point at infinity of lR2n +2 not belonging to ]Pl. Letf1Y be the projection of/oY into lRl made from Po, so that the direction of the projection is not parallel to lR l . For the imbeddingf1 of Vinto lRl to be locally regular, it is sufficient that no tangent of fo V pass through Po and, for it to be injective, it is sufficient that no chord offo V pass through Po. By Lemma 2, we can satisfy these conditions by choosing Po as far as we wish, in ]p2n +1, from the point at infinity of the lines oflR2n +2 orthogonal to lR2n+1. We can then establish thatf1 is a regular imbedding of V into lRI' such that the distance between the orthogonal projections ofhY and of/oY in lR 2n +1 (the latter reduces tofy) is less than 82- 1 . Further, we note that as the direction of the projection from Po is parallel to lR2n +2, iffoYElR2n+1+k(k~O), we haveflYElR 2n + 1+k.

14

Chapter I. Notions About Manifolds

Similarly, by starting with/1 instead offo, we can obtain a regular imbedding /z of Vinto IR~ such that the distance between the orthogonal projections of/1y and /zy in 1R2n +1 is less than e2- 2 and such that if I1YEIR 2n+1+k, then, 12YEIR2n+1+k. The point /zy is determined from I1Y by projection from a conveniently chosen point at infinity. By continuing in the same manner, we obtain, for each integer h > 0, a regular imbedding Ji, of V into 1Rr;' such that the distance between the orthogonal projections of Ji,y and Ji,-lY in 1R2n +1 is less than 8r h and such that, if loYEIR2n+1+k, then,./hYEIR2n+1+k. The pointJi,y is determined from./h-1Y by a projection from a conveniently chosen point at infinity. Now, because of the assumed properties of I: for each yE V, there exists an integer k, depending on y, such that IYEIR 2n+1+k; thus, IkYEIR 2n +1+k n 1R~=1R2n+1 and.hY=.h+1Y=Ik+2Y= .... The imbedding g of V into 1R2n +1, defined by gy=.hy, has all the required properties. In fact, in a neighbourhood of each point of V, we have gy = Ji,y for a sufficiently large number h depending only on this neighbourhood, so that asJi, is locally regular, g is also. Then, if y=l=z, as gy=Ji,y and gz=Ji,z for a common sufficiently large number h,Ji,y =l=Ji,z implies gy =l= gz and so, g is injective. Finally, the distance ofly to gy, that is, the distance between the orthogonal projections of loY and of.hY into 1R2n +1, is less than e(r1 +r 2+ ... +2- k) < e. Thus, Theorem 5 is proved. D Let us consider such a proper, regular imbedding g on V into IR2n +1. Let % be a (2n + 1) dimensional manifold consisting of all the pairs (x, e) where x is a point ofgVand is a vector in 1R2n +1 with origin atx and normal togVatx. Let Jl be the map Jl (x, e) = x + eof % into IR2n +1, associating with (x, e) the endpoint of the vector We easily verify that this map has non-vanishing Jacobian at (x, 0). Using the implicit function theorem, we know that there exists a strictly positive function Q(Y) defined on Vand which may be supposed Coo, such that, in the domain %1 C % consisting of the pairs (x=gy, e) such that lei < Q(y), the Jacobian of Jl does not vanish and the map Jl restricted to %1 has no critical points and is a local homeomorphism. By noting that g is proper and injective, we easily verify that we may choose the function Q(Y) so that the map restricted to %1 is also globally injective. Thus, D=Jl%l is a neighbourhood of gV in 1R2n +1 called a tubular neighbourhood. Let Ny be the set of points in the plane normal to g V at x = gy for which the distance to x is < Q(y). As y moves over V, Ny sweeps out D and in this way, D is seen as a space fibred by the Ny. The map p of D onto V such that pz = y if ZENy is the projection of this fibration onto its "base space" V. The map Pt of D into itself, defined for O~ t~ 1 by Ptz=z+ t(gpz -z), is the identity for t=O and gp for t = 1 and provides a retraction of D onto g Vin which each fibre Ny retracts onto its centre gy.

e

e.

Chapter II. Differential Forms

§4. Differential Forms of Even Type On a manifold, we obtain differential forms of degree 1 as sums of the products of a function g by the differential df of another function f,

Expressing this in terms of the local coordinates Xl, ... ,xn , the above differential form reduces to the expression n.

L ai dx'

i=l

with ai=

8f

L g -8 X

i'

If we change the local coordinate system, the coefficients ai transform as the components of a covector. By definition, the form is zero at a point if all its coefficients vanish at this point, and two forms are identical if and only if their difference is zero at each point. A form will be said to be C if its coefficients are cr. From forms of degree 1, we obtain exterior differential forms of higher degree (which in general we will simply call forms, provided no confusion can arise) by using the operation called exterior multiplication, combined with additionl. This operation, which is represented by the symbol /\, obeys the usual associative and distributive laws together with the following pseudo-commutativity laws:

dx i /\ dxi = - dx i /\ dx i, dx i /\ dXi = 0 a /\ dXi =dXi /\ a=adx i dXi /\ adx i = adx i /\ dx i , where a is a scalar. From these laws, an exterior differential form rx of degree p (O;;i;p;;i; n), expressed in local coordinates, reduces to the form rx=

L

rxil ... ipdxil/\ ... /\ dx ip

it < ... n reduce to zero. A form of degree 0 is simply a function. The form IX is said to be zero at a point if all the coefficients rlil ... ip of its local expression vanish at this point, and two forms are equal if and only if their difference is zero at each point. The form IX is also said to be C' if its coefficients are C. The support of a CO form is the smallest closed set outside of which the form is zero. It is clear that the associative, distributive and pseudo-commutative laws are independent of the coordinate system. If rl and f3 are two forms of degrees p and q respectively, we have The coefficients lXil .. . i p of the form IX, relative to local coordinates, are defined for il < ... < ip • We agree to define them for all system of indices by the antisymmetry condition, so that rlh ... jp = 0 unless the indices}l,' .. ,}p are all distinct, and IXh ... jp= ±rlit ... i p if}l,'" ,}p is a permutation ofi l , ... , ip, with the + sign for an even permutation and the - sign for an odd permutation. Then, we can write 1

rl=-

p!

where, in the summation, each index varies from 1 to n. If Xl, ... ,xn is another local coordinate system, we have · dXl=

I -oxi d-]' x GX j

,

and by noting the preceding laws, 1

rl=, .

with (1 )

p.

, ,d-h L... rlh ... jp x

1\ . . . 1\

d-X j p

] l ..... ]p

fl..] l ... ]p . =

" L..

In a change oflocal coordinates, we see that the coefficients of a form of degree p transform according to the same law as the components of a covector, or anti symmetric covariant tensor of rank p. A form of degree p also defines at each point of Va p-covector. Alternatively, prescribing a p-covector at each point of V defines a form of degree p which we 2 It suffices to verify that the local expressions, when we define their product by the formula (2) below, gives rise to an algebra satisfying the above laws.

§4. Differential Forms of Even Type

17

can always represent by a sum of terms of the form

where g is a function with compact support and the Ji are Coo functions, and where the sum has a finite number of terms if the support of the form is compact and an infinite but locally finite number otherwise. This is easily shown by using a partition of unity. The forms defined here will be called ofeven type or even, so that they may be distinguished from the odd forms which will be introduced in the following section. The differential of the form a of degree p, which we suppose C 1 , is the form da of degree (p + 1), defined in the domain of a coordinate system using the local expression for a by putting

L

da=

dait ... ip /\ dx i ,/\

••• /\

dx ip

il< ... 0 remains bounded, the forms

remain in a set bounded to order q; thus, the same is true for the forms R*¢-¢ A*¢ --'------'- and - e

e

and this implies property (6). Property 5) also follows from the preceding by noting that if we replace L(Y) by L( -y), R* and A*w are replaced by Rand A. This implies property 4). The formula in 1) is deduced from the homotopy as in the case of Proposition 1. Let F be the set swept out when the support of ¢ is acted upon by all the transformation s~ with 1111 < e. The formulas of definition (4) show that the supports of R*¢ and A*¢ are contained in F. So, RT[¢]=AT[¢]=O if the support of T does not meet For, what is essentially the same, if the support of ¢ does not meet E(T,e). This means that the supports ofRTand ATare contained in E(T,e). Moreover, if the support of ¢ is outside of IBn, the homotopy St~ reduces to the identity outside oflBn,S~¢ =0, and so, A *¢ =0 and AT[¢] =0. AT

Chapter III. Currents

68

is thus zero outside of IBn and its support is thus contained in E(T, 8) (} IBn. This proves 2'). If Tis C', RTis C', as is AT. If Tis C' in a neighbourhood of a frontier point Xo of IBn, we can decompose Tinto the sum T= Tl + T2 of two currents where Tl is C' everywhere and where T2 vanishes in a neighbourhood of Xo. Thus, RTI is C' (everywhere) and RT2 vanishes in a neighbourhood of xo, so that RT is C'in a neighbourhood of Xo. This is true for 0 ~ r ~ 00 and 3') follows. We can express this property by saying that R is regularizing in IBn and is nowhere deregularizing. 0 If the form, satisfies, (1]) = r ( -1]), we have, as in the case of Proposition 1, R=R* and A=A*w. Further let us note that the operators Rand b commute, just as moreover do R and b. In fact, if we replace T by bT in formula 1), it follows that RbT-bT = bAbT, and if we take the boundary of each member of this formula, we obtain bRT-bT=bAbT. So, it follows that RbT=bRT. Theorem 12. On a manifold V, we can construct linear operators R and A, depending on positive parameters 8 1 , 82' . . . which are finite or infinite in number according as V is compact or not, which have the following properties: 1) If T is a homogeneous p dimensional current in V, RT and AT are respectively homogeneous p and (p + 1) dimensional currents which have the same parity and which satisfy

RT-T=bAT+AbT. 2) The supports of RT and AT are contained in any given neighbourhood of the support of T provided that the parameters Gi are sufficiently small. 3) RT is CX); if T is C', AT is C'. 4) If ¢ varies in a set bounded to order q and if each Gi varies in such a way that it remains bounded above, R¢ and A¢ remain in a set bounded to order q. 5) Ifeachparameter 8i tends to zero, RT[¢l~T[¢l and AT[¢l~O uniformly on each bounded set offorms ¢; if T is continuous to order q, the convergence is uniform on each set of forms which is bounded to order (q + 1).

In order to construct the operators R and A, we commence with a locally finite covering {Vi} of V such that each Vi is homeomorphic to the ball IBn via a homeomorphism hi which is Coo and which can be extended to neighbourhoods of Vi and of IBn. Using these homeomorphisms, the operators R and A which are defined in IR n can be transported to V. In fact, letfbe a CX) function, ;;; 0, which has its support in the neighbourhood of Vi and which is equal to 1 in another smaller neighbourhood of Vi. If T is a current in V, T' = fTis a current which has its support contained in a neighbourhood of Vi and hiT' is a current which has its support contained in the neighbourhood of IBn. The support of Til = T - T' does not meet Vi. By replacing the parameter G occuring in R and A with Gi and putting

69

§ 15. Regularization

we define operators Ri and Ai which possess properties in V corresponding exactly to those of the operators R and A in JR.". Here Ui plays the role ofIB" and 8i replaces 8. Put In the neighbourhood of each compact set K, the operators Rh reduce to the identity and the Ah reduce to zero whenever h is sufficiently large because Uhdoes not meet K. It follows that R= lim R(h) and A= h~oo

L 00

h=1

A(h)

are well defined operators, and there exists an integer ho depending only on the compact set K such that RT[4>]=R(h)T[4>] and AT[4>]=

h

L

i=1

A(i)T[4>]

for h ~ ho, and for all currents Tand all forms 4> with support in K. RTand ATare thus well defined currents. By virtue of the properties of Rh and Ah which correspond to part 1) of Proposition 2, we have

and by adding term by term these relations for h = 1,2, ... , it follows that RT-T=bAT+AbT.

This proves 1). 2) follows from the corresponding properties of Ri and Ai. If the support of T is contained in an open set U, we can successively determine bounds for each of the parameters 8i (i = 1, 2, ... ) so that, as long as these bounds are not exceeded, the supports of R(h)T and A (h)T remain in U. Thus, the same will be true for the supports of RT and AT. The fact that RT is Coo follows since R(h)T is Coo in Ui for h ~ i. In fact, by virtue of Proposition 2 (part 3'), Ri regularizes in Ui and is not deregularizing anywhere. The fact that AT is Cr if Tis C' follows from the corresponding properties for the operators Ri and Ai, and 4) follows in the same way. To show 5), note that we have

L 00

RT[4>]-T[4>]=

h=1

(R(h)-R(h-1»)T[4>],

00

AT[4>]=

L

h=1

A(h)T[4>].

Chapter III. Currents

70

If ¢ remains in a bounded set, there exists an integer ho which depends only on this set such that the general term of each of these series vanishes for h > ho . Thus, it is sufficient to prove that, as ch ->0,

uniformly on each bounded set of forms (bounded to order (q+1) if Tis continuous to order q) and uniformly with respect to Ci (i < h) which we also suppose bounded. Now, we have (R(h) - R(h-l)) T[¢] = (RIrT - T) [R(Ir-l)*¢], A (Ir)T[¢] =AhT[R(h-l)* ¢],

where R(h-l)*¢ = Ri1'- l Ri1'-z ... R{¢ remains in a set bounded to order(q+ 1) if ¢ remains in a set bounded to the same order and if the Ci (i < h) remain bounded. Thus, our assertion follows from the corresponding properties of Rh and Ah (Proposition 1, part 6). In general, in a manifold V, we shall call each operator R which depends upon positive parameters Ci, to which is associated an operator A, and which possesses the properties expressed in Theorem 12 a regulator. The operators defined above have the transposes (jJ

R*= lim R(h)* and A*= h~(jJ

I

A (h)*,

h=l

where

Even if R* = R;, we do not have R* = R. Nevertheless, the operators R* and A *w possess all the properties of R and A expressed in Theorem 12: R * is a regulator and A *w is its associated operator. We verify that R' = RR* and A'=t(A +AR*+RA*w+A*w)

possess the same properties and furthermore that R' =R'* and A' =A'*w. Thus: There exist operators R and A having properties 1) to 5) of Theorem 12 and such that R*=R and A=A*w.

§ 16. Operators Associated with a Double Current 12 Let L(x,y) be a double current on Vx W, ¢ = ¢(x) a form in qyv and Ij; = ~J(y) a form in qyw. 12

For this §, see Schwartz [4].

§16. Operators Associated with a Double Current

71

Considering ¢ as fixed and ljI as variable, the expression S S ¢(x) /\ L(x,y) /\ ljI(y)=wxL(x,y)[¢(x)ljI(y)] x y

is a linear functional of ljI which is continuous in f0 w . Thus, it is a current on W and we will denote it by A¢. Considering ljI as fixed and ¢ as variable, this same expression is a linear functional of ¢ which is continuous in f0 y . Thus, it is a current on Vand we will denote it by A *ljI. We have A¢[ljI]=¢[A*ljI] = S S ¢(x) /\ L(x,y) /\ ljI(y) x y

and we write A¢=S ¢(x)/\L(x,y),

A*ljI=S L(x,y)I\ljI(y).

x

y

The operator A thus defined is a continuous linear map of f0 y into f0i,v. In fact, the continuity of the double current L implies that A¢ remains in a bounded subset of f0i,v whenever ¢ remains in a bounded subset of f0 y . The operator A * which we will call the transpose operator of A, or moreover, the topological transpose of A in order to distinguish it from the metric transpose operators which we define later on, is a continuous linear map of f0 w into f0 We say that the double current L(x,y) is the kernel of the operator A. If A¢=O for all ¢Ef0 y , we have wxL(x,y)[¢(xhif(y)] =0 for all ¢Ef0 y and ljIEf0 w ; by virtue of Theorem 6, this implies that the double current L(x, y) vanishes. It follows that different double currents are the kernels of different operators!3 . We verify that if L(x,y) is the kernel of A, the operators bA, Ab, wA, . .. which occur on the first line of the table below, have kernels which are double currents occurring on the second line of the same table:

v.

A

bA

Ab

wA

Aw

wA

Aw

We look for the transpose A** of A*. As T[¢]=¢[wT], we have A*ljI[¢] =w¢ [A*ljI] =Aw¢ [ljI] = !/t[wAw¢]

and it follows that A**=wAw. 13 From a theorem announced by Schwartz ([4], Theorem 1), which however we do not use here, we have, conversely, that to each continuous map of i0 v into i0iv is associated a double current on Vx W which is the kernel of this map.

Chapter III. Currents

72

We see that the second transpose of A is the transformation of A by W. For A ** = A, it is necessary and sufficient that A commutes with W. In particular, this is so whenever A preserves both dimension and degree. The kernel of A* is wyL(y,x) and that of A**=wAw is wxwyL(x,y). Let 11 and 12 be Coo maps of the same manifold Z into V and W, and let 1= (ft ,fi) be the resulting map of Z into V x W. If S is a current on Z whose support intersects the inverse imager 1K of each compact set K C V x W along a compact set, JS is a well defined double current according to the definition of § 13, provided that if S is not odd at x (resp., y),Jl (resp.,fi) is oriented. We look for the operator of the kernelJS. We have

Jx J O, and we consider those for which k has its maximum value k o . Let ai /\ IXi be one of these. As the support of bT does not contain any interior point of ai, we must have ai /\ blXi = O. If we denote by I the injection of ai into V, this signifies that I*IXi is closed on ai' From the lemma established at the end of § 19, there then exists a form Wi in V which, like lXi, is Coo and is such that I*(bwi - IXi) = 0 on ai' This is equivalent to ai /\ IXi = ai /\ bWi. Denote by Tl the sum of all ai /\ Wi corresponding in this way to terms ai /\ IXi of Toftype (q+ko,ko). Tl is a current of S. bTl and Tthen have the same terms of type (q+ko, k o), and so, Tis homologous to a current T' = T-bTl which is a current of S which contains only terms of type (q+k,k) with k 1, we proceed by induction on q -po Then, by hypothesis, we have

and so,

Since in this case J1~)(bai,f)=O, formula (4) gives

and this immediately gives formula (3). We have seen that the support of (c,f) is contained in the support of c. We will now look for the relation between this support and the cochainf We call the open support ofJthe set in Vobtained from the union of the interiors of all the cells on which J does not vanish and all the cells which have in their frontier a cell on whichJdoes not vanish. This set is clearly open; if the degree ofJis p, it does not contain any cell of dimension < p. Thus, we have the following proposition: The support oj (c,f) is contained in the open support ojf It is sufficient to prove this for (ai,f) and we can suppose that J is homogeneous. Further, by taking the dimension of ai to be q and the degree ofJ to be p, this follows from 2 b) if q - p = 0, and, if q - p > 0, this follows from 2 c) by induction on q-p. We denote by /; the cochain of S which takes values + 1 on ai, -1 on - ai and on all other oriented cells of S, and we put

°

§22. Duality in a Manifold Endowed with a Polyhedral Subdivision

93

where 1 denotes the even n dimensional chain of S which, considered as a current, is equal to the function 1. If q is the dimension of ai ,j; has degree q and the chain ei has dimension p = n - q and is odd. Formula (3) shows that bei is a linear combination of the ej, since dj; is evidently (being cochain of S) a linear combination of the jj. We will show that the Kronecker indexei /\ aj[1] is always defined and has the value if i"r-j and 1 if i=j,

°

(5) From our definitions, it is clear that this symbol is equal to zero whenever the sum of the dimensions of ei and aj does not equal n, that is, whenever ai and aj do not have the same dimension. Thus we can confine ourselves to examining the case where ai and ajhave the same dimension q. In this case, the dimension of ei is

p=n-q. The support of ei is contained in the open support of j; which consists of the interior points of the cell ai and all the cells which have ai in their frontier. Thus it does not contain any point of any cell ak of dimension ~ q other than ai. This implies that

This however also shows that the chains bai and ei, as well as the chains ai and bei, do not meet. This ensures that the symbol ei /\ a;[l] is defined. To prove that it equals 1, we will proceed by induction on p. If p = 0, ai has dimension n and in the expression 1 = Ckak for the chain 1 of S, its coefficient Ci is 1. We have

L:

ei=(l,j;)=

L: ck(ak.j;)Oi=CiOi. k

But ai, considered as a current, is a function equal to Ci in the interior of the cell ai and zero outside; 0i/\ai[l]=Oi[a;], which is the value of this function at the point 0;, is equal to Ci. Hence, ei/\ai[l]=cr=1. If p > 0, we have q < n and we can find a cell ah of dimension (q + 1) which has ai on its frontier. The coefficient of ai in the expression for bah equals ± 1 and, without loss of generality, we can suppose this coefficient equals 1. Then, bah=ai+' .. and (6) However we also have bah· j; = 1, and so, ah· dj; = 1. This shows that, in the expression for dj; as a linear combination of .h (each odd cochainf is equal to such a linear combination, namely, f = (ak -/)ak), the coefficient of fh is equal to 1,

L:

dj;=fh+··· and (1At;)=eh+ ....

94

Chapter IV. Homologies

Noting formula (3) and that the difference between the dimension n of the chain 1 and the degree q off; is n-q-p, we obtain

and it follows that

However, by virtue of the properties of the Kronecker index (property d of § 20),

since dah = wbah = ( -1)P bah, where bah is of degree p, and so

By composing with (6), it follows that ei A ai[l]=eh Aah[l]. As the dimension of eh is p -1, it follows, by induction of p, that ei A ai[l] = 1. For any odd cochainf of S, (1,f) is a linear combination of the ei' Thus, for each even chain c of S, the Kronecker index (1,f) A c [1] is defined, provided that for c is finite, and (l,f)Ac[l]=cf

This formula is evidently still true in the case where f and c have the same parity, each side then being equal to zero. To compare the proof, it is still necessary to establish this in the case where f is even and c is odd. To do this, we remark that the open support of fj is an orientable domain which forms a neighbourhood of ei and to which we can extend each orientation Gj of a neighbourhood of aj. Denote by fj the even cochain of S which takes the value + 1 on Gjaj, -ion - Gjaj and 0 on all other oriented cells of S. Proceeding by induction on the dimension of (c,f1) and noting (2b) and (2c), we see that (c,fj) = Gj(c,fj) for each chain c of S. In particular (l,fj) = Gjej and so, for each even cochain f of S, (1,f) is a combination of Gjej. Since each odd chain c of S is a combination of the Giai, the formula (1,f) A c[1] = c -f follows from the fact that Gjej A Giai [1] = 61 and this fact is essentially what was to be established. We have thus proved the following proposition.

Proposition 3. In a manifold V endowed with a polyhedral subdivision S, we can associate to each cochain f of S a chain (1,f) in V such that b(l,f) = -w*(l,df) and (1,f) Ac[l]=c·j, for all chains c of S, the Kronecker index always being defined if c or f is finite.

§22. Duality in a Manifold Endowed with a Polyhedral Subdivision

95

This proposition, in conjunction with Propositions 1 and 2, will enable us to easily establish the duality theorem in the case of a manifold endowed with a polyhedral subdivision. The statement of this theorem is the following.

Theorem 17. In a manifold V,Jor a closed current Tl to be homologous to zero, it is necessary and sufficient that T2 /\ Td 1] = for each closed current T2 with compact support. For a closed current Tl with compact support to be compactly homologous to zero, it is necessary and sufficient that T2 /\ Tl [1] = for each closed current T2 .

°

°

The necessity of the condition stated in each of the parts of this theorem has been established in § 20 (property d), where we have shown (property 3) that the symbol T2/\ Tl [1] is always defined when the two currents Tl and T2 are closed and one of them has compact support. It follows that if T2 , for example, has compact support, the value of T2 /\ Tl [1] depends only on the homology classes of He(V) and H(V) containing T2 and Tl respectively. To prove the sufficiency of the condition stated in the first part, suppose that Tl is closed and non-homologous to zero. From Theorems 14 and 16, Tl is homologous to a chain c of the polyhedral subdivision S of V. As this chain is not homologou& to zero, it does not bound a chain of S, and by virtue of Proposition 2, there exists a finite cocycle f of S which does not vanish on it, c1*O. From Proposition 3, T2 =(1,f) is thus a closed current with compact support and T2 /\Td1]=cf*0. The proof of the sufficiency of the condition stated in the second part is exactly analogous: it is sufficient to use Proposition 1 instead of Proposition 2. D We remark that if, in the statement of this theorem, we suppose that the currents Tl and T2 are homogeneous, we obtain an equivalent statement. In fact this follows from the following observation. For a current to possess one of the properties, namely, to be closed, with compact support, homologous to zero or compactly homologous to zero, it is necessary and sufficient that each of its homogeneous components possess the same property. We can thus interchange the order of Tl and T2 in the Kronecker symbol, since in this case Tl /\ T2 [1] = ± T2 /\ Td1]. Noting Theorems 14, we see that Theorem 17 is equivalent to the following.

Theorem 17'. For the current T to be homologous to zero, it is necessary and sufficient that T[ 4> ] = for all closed Coo forms 4> with compact support. For a current Twith compact support to be compactly homologous to zero, it is necessary and sufficient that T[ 4> ] = for all closed Coo forms 4>.

°

°

Noting Theorems 14 and 16, we again obtain an equivalent statement by considering, instead of a form 4>, a chain c and by replacing T[ 4> ] with T /\ c [1]. In § 23, we will extend the proof of this theorem to the case of any manifold without having to suppose that it is endowed with a polyhedral subdivision. There, we will furthermore complete Theorem 16 as it is stated by establishing the following proposition.

Chapter IV. Homologies

96

In a manifold endowed with a polyhedral subdivision S, if a chain of S bounds a current, it bounds a chain of S; if it bounds a current with compact support, it bounds a finite chain of S.

In fact, if the chain c of S is homologous to zero, we have (l,f)/\c[l]=O

for each finite cocycle of S, as then (1,f) is a closed current with compact support. Thus, c .f = 0, and by virtue of Proposition 2, c bounds a chain of S. If c bounds a current with compact support, it is finite and similarly we see that c -J= for any cocycle f of S. Thus, by Proposition 1, it follows that c bounds a finite chain of S.

°

§ 23. Duality in Any Differentiable Manifold To extend Theorem 17 to any differentiable manifold V, without having to suppose that it possesses a polyhedral subdivisions, we will make use of Theorem 5 (Whitney's Theorem), which affords the existence of a proper, regular embedding g of V into IR.2n+l, X =gy, yE V, xEIR.2n+l. We consider the tubular neighbourhood D of g V in IR.2n + I defined at the end of § 3. As for any open set in a Euclidean space, D can be endowed with a polyhedral subdivision by means of the following well known procedure. Let Iffm be the set of cubes in IR.2n + I defined by

where Xl, . . . , X2n + I are cartesian coordinates in IR. 2 n + 1 and where hi are any integers. For any integer m ~ 0, we take the cubes of Iffm which are interior to D and which (for m > 0) are not contained in a cube of Iffj interior to D (0 o. If ¢ is a Coo form with compact support, A(y)=S ¢(x)/\L(x,y) x

§27. Properties of the Geodesic Distance

119

is a Ceo form. Moreover, if ¢ varies so that it remains bounded to order 0, A¢ also remains continuous in a neighbourhood of each point and locally bounded to order O.

In fact, we decompose the integral defining A¢(y) into the sum of two integrals, taken over a neighbourhood D' of a point Yo and its complement, respectively. Let A' ¢(y) and A" ¢(y) be the values of these integrals so that we have A¢(y)=A'¢(y)+A"¢(y). As D' shrinks to a point, the coefficients of A' ¢ (y) tend to zero, uniformly with respect to y in a neighbourhood of any point, in particular, Yo, and uniformly with respect to ¢ on each set bounded to order zero. As the coefficients of A" ¢ are continuous at the point Yo and also, continuous even when ¢ varies so that it remains in a set bounded to order zero, it is the same for those of A¢ (y) and this proves the assertion concerning the continuity. On the other hand, the fact that A¢ remains locally bounded to order zero is immediate. 0 Lemma 6. Let L 1(x,y) and L 2(x,y) be two double forms on Vx V which are continuous away from the diagonal and are such that L1 (x, y) = 0 (rei - n) and L 2(x,y) = O(r e2 - n) with e1 > 0 and e2 > O. If P denotes the projection P(x,y,z)=(x,y) of Vx Vx V onto Vx V, let the support of the triple form L1 (x, z) 1\ L 2(z, y) in Vx Vx V intersect the inverse image p- 1K of each compact set K C V x V in a compact set. Then, the double form

on V x V is continuous away from the diagonal. Morevoer, if e1 + e2 -n > 0, it is continuous even on the diagonal, whereas L(x,y)=O(rel+e,-n) ife1 +e2 -n0. To study L(x,y) away from the diagonal, we consider two domains D' and D" which have no common point in V and let D '" be the complement of their union. If L'(x,y), L"(x,y) and L"'(x,y) are the values of the integral taken over D', D" and D'" respectively, we have L(x,y)=L'(x,y) + L"(x,y) + L'''(x,y). For xED' andYED", by virtue of Lemma 5, L'(x,y) and L"(x,y) are continuous and it is immediate that L"'(x,y) is also continuous. The implies the continuity of L(x,y) away from the diagonal. To study L(x, y) in a neighbourhood of the diagonal, we consider a point vof V and the normal coordinates relative to this point which we will suppose are orthonormal at v. The domain of this system contains a geodesic ball Se with centre v and radius (l. Put Le(x,y)=

S L 1(x,z)I\L2(z,y). zeSe

Chapter V. Harmonic Forms

120

It is immediate that L(x,y) -LII(x,y) is continuous for XESII andYESII • Hence, it suffices to study Lg{x,y). Denote by Xi, yi and Zi (i = 1, ... , n) the coordinates of X, y and z in the system considered, and by d(x,y) the Euclidean distance

When X and y remain in SII' the ratio of r(x,y) to d(x,y) lies between two fixed positive numbers. Thus, a function of two variable points in SII which is O(rk) is O(dk) and vice versa. Each coefficient of LII(x, y) in SII x SII is then the sum of a finite number of terms of the form

J

Fl (x, z)F2 (z,y)d el - n(x, z)d e2 - n(y, z)dz1 . .. dz n,

zeSe

where F1(x,z) and F 2 (z,y) are bounded functions. Thus, its coefficients are bounded above, up to an almost constant factor, by All =

J del-n(x, z)de2 - n(y, z)dz1 . .. dzn. So

To establish the assertion of Lemma 6 in the case where el + e2 - n < 0, it thus suffices to show that AII(X,y)=O(del+e2-n). To this end, we partition the domain of integration into two parts, one consisting of the points in the ball D' centred at x and of radius 2d(x,y), the other consisting of the part D" of SII outside this ball. Let All = A~ + A; be the corresponding decomposition of All. A~ is bounded above by the integral taken over all the ball D' which has the value, up to an almost constant factor, of del+e2-n(x,y). We immediately see this by transforming the integral by a homothety with centre x and ratio d-1(x, y). Thus, we have

f D' h . d(y,z) .. I . OUtS1·de o N ext, wh en z vanes , t e ratlO d(x, z) attams 1ts east upper

bound and greatest lower bound when z is at the ends of the diameter of D' passing through x and y. These bounds are and t. On the other hand, D" is contained in the domain B bounded by the Euclidean spheres centred at x and of radii 2d(x,y) and 2(l (we suppose x andy are in SII). It follows that A; is bounded above, up to an almost constant factor, by

t

The integral is equal, up to an almost numerical factor, to (lei +e2- n _del +e2 -n(x,y)

el +e2- n

§28. The Parametrix

121

If e1+ez-nO. Consider the integrals LR(x,y) and AR(x,y), analogous to Le(x,y) and AeCx,y) but defined over SR instead of Se' where we suppose R < Q. Divide the integral AR into two integrals defined over the two parts of SR where d(x,z)~d(y,z) and d(x,z)~d(y,z) respectively. We bound these integrals above by replacing in one dey, z) by d(x, z) and in the other d(x, z) by dey, z), and we again bound them above by then extending the field of integration to all of SR' Thus, AR
O Sn

This completes the proof.

S *yl =¢(x). D,

D

§29. The Regularity of Harmonic Currents

127

§ 29. The Regularity of Harmonic Currents Theorem 21. In a Riemannian space V, each set ofcurrents Twhich is bounded in f!fi' and such that the set of LI T is bounded in $ is bounded in $.

To say that a set of currents is bounded in $ implies that this set is contained in $, that is, all currents of the set are equal to Coo forms. Before proving this theorem9 , we will show two consequences.

Corollary 1. A current T such that LI T is Coo in a domain D of V, in particular, each current harmonic in D, is Coo in D. In fact, it is sufficient to apply the above theorem to the Riemannian space D formed by D and to a set reduced to a single current.

Corollary 2. If a sequence of currents Th (h = 1,2, ... ) is such that,for h-+ 00, Th-+O in f!fi' and LlTh-+O in $, then, Th-+O in $. In fact, from §§ 9 and 10, the hypothesis implies the existence of a sequence of numbers mh-+oo such that mhTh remains bounded in f!fi' and mhLlTh=LI(mhTh) remains bounded in $. Theorem 21 shows then that mhTh remains bounded in $ and so, it follows that Th-+O in $. D Proof(ofTheorem 21)10. Let go be a set of currents such that the set of LiTis

bounded in $. Then A)

If f!4

is bounded in f!fi'P+l, go is bounded in f!fi'p.

In fact, this follows immediately from formula (I) of Lemma (3, § 28), which may be written as T=QLlT+Q'T,

and from Lemma (2, § 28). In the same way, we see that: B)

If f!4

is bounded in $ P, go is bounded in $

P +1 .

Furthermore, we show that: C)

If f!4

is bounded in f!fi'o, go is bounded in $0.

By using the formula

which reduces to (I) for m = 1, and which we establish by induction on m by substituting for T, in the last term of the second member, the value given by (I), 9 Concerning this theorem and the two corollaries, see Schwartz [2], pp. 136-143, de Rham [10], pp. 36-39 and 61. See also Kodaira [2], pp. 608-626. 10 The principle of the following proof was communicated to me by L. Schwartz.

Chapter V. Harmonic Forms

128

we see that, to establish C), it is sufficient to prove that if T remains bounded in ~/O, then Q,mT remains bounded in 18°, provided m is sufficiently large.

Let qm(x,y) be the metric kernel on Qm. We have ql(X,y)=q(x,y) and

z

The metric kernel of Q,m is q;"(X,y) = qm(y, x). Lemma (2, §27) shows that, if 2m -n0; and if 2m -n > 0, qm(x,y) is everywhere continuous (even on the diagonal). Thus, we suppose that m>~ so that qm(x,y) is continuous. The topological kernel L(x,y) of Q,m, which is simply *xqm(x,y), is then a continuous double form. By considering it as a form iny, its coefficients are forms in x belonging to ~o and depending continuously on y, that is, they are continuous functions of y with values in ~o. If TE~/O, we obtain the form Q,mT(y) by applying the operator T to its coefficients. Thus, the coefficients of Q 1m T(y) are continuous functions of y (with real values) which remain bounded whenever y remains in the neighbourhood of a point and T in a bounded subset of ~ '0. This is precisely what we had to prove. Thus, let us consider a set of currents T which is bounded in ~' and such that the set of AT is bounded in 18. Let D be a domain with compact closure D, and let 1/1 be a Coo function with compact support which is equal to 1 inDo The set of currents 1/1 Tis bounded in 18 I, since the set of T is bounded in ~ By virtue of Theorem 8 (§ 10), this set is bounded in 18 IS, for s sufficiently large. It follows that the set of currents T restricted to D is bounded in ~D . Proposition A), applied s times consecutively by considering the Riemannian space formed by D, shows that this set is bounded in ~DO. Proposition C) then shows that it is bounded in tfg, and proposition B) shows finally that it is bounded in 181) for any p, that is, it is bounded in 18tJ. This implies that the set of currents T is bounded in 18. This is 0 what we wished to prove. I.

The same method applies to other differential operators. For instance, consider the operator A + c, where c is a constant or a Coo function. Formula (I) of lemma (3, § 28) can be written as T=Q(A +C)T+(Q' -Qc)T

and the same argument as above shows that, ifa set ofcurrents T is bounded in ~ and such that the set of (A +c)T is bounded in 18, it is bounded in 18. As the set of Tis always bounded in ~/, if we suppose only that the set of (A +c)T is bounded in tfP (resp., ~/P+l), we can conclude that the set of Tis bounded in tf p +1 (resp., ~/P). I

§30. The Local Study of the Equation LJjl = {3. Elementary Kernels

129

§ 30. The Local Study of the Equation .1p = /3. Elementary Kernels Let us propose the following: Given a form f3 in V, find a form Jl satisfying the equation LJJl=f3 in a domain DC V. We will show that, if D is sufficiently small, this problem is always solvable ll . Let f be the characteristic function of D and denote by Q l and Ql the operators obtained by multiplying Q and Q on the right by the operator consisting of multiplication by f We have

J W(X,y)A*e(y)

Ql(e)=Q(fe)=

yeD

and an entirely analogous expression for Ql' Formula (II) of Lemma (3, § 28) implies that

e

As this is so, for Jl = Q l to satisfy the equation LJJl = f3 in D, it is necessary and sufficient that satisfy the integral equation

e

Now, if D is sufficiently small, this equation can be solved by the LiouvilleNeumann method, based on the identity

In fact, suppose that D is contained in the domain of a coordinate system, and denote by 14> 1the least upper bound of the absolute value of the coefficients of the form 4>inD. We have an inequality of the form IQl4>1 ~kl4>l, wherekis a number independent of 4> and depends only on D. Also, if D is sufficiently small, k < 1. Thus, 1Qi"f3 1~kmlf3l, and the series

e= I

00

Qi"f3

m=O

e

converges uniformly and provides a solution of the integral equation. If we denote by qm(x,y) the metric kernel of the operator Qi" (which we will avoid confusing with the kernel of Qm, denoted by the same symbol in §29), the metric kernel of the operator

r

00

=

I

o

Q l Qi" which, applied to f3, provides the

11 The principle of the method applied here is due to E. E. Levi [2]. See de Rham [10], pp. 60-61. For analytic Riemannian spaces, the existence of an elementary kernel has been established by Kodaira [2], pp. 612-618, by a method of Hadamard.

Chapter V. Harmonic Forms

130

solution /1 = Q 1 ( of the equation ,1/1 = /3, and which we will call the elementary kernel relative to the operator ,1 and to the domain D, is given in D x D by 00

L S W(X,Z)A*zqm(Z,y).

y(x,y)=w(x,y)+

m=l

zeD

It follows from this, noting Lemma (6, § 27), that y(x,y)=O(?-'), and more precisely, if n>4, y(x,y) =w(x,y) + 0(r4 -.).

We can extract from this a proof of Theorem 21, a little different from that given above. We will restrict ourselves to indicating the idea. First, with the aid of the above expression and by using Lemmas (4 and 6, §27), we show that, in D x D, y(x,y) is Coo away from the diagonal. Next, by introducing a function a(x,y) defined in D x D like the function a(x,y) of § 28, we show that the operator r 1 of the metric kernel }'l (x,y)

= a(x,y)y(x, y)

and its metric transpose r; send each bounded set in is'D into a bounded set in is'D' The fact that ,1r=l implies that ,1xy(x,y)=O for x=l=y and the form Y2(X,y) which is equal to ,1xl'l(X,y) away from the diagonal and zero on the diagonal is Coo in D x D. If r 2 is the operator of the metric kernel }'2(X,y), we have ,1r1 = 1 + r z and thus, by transposing the operators, r;,1' = 1 + r~. As the operator r~ is regularising, the relation T= r;,1 T - r~ Tthen immediately shows that, if T remains bounded in f0 i> and if ,1 T remains bounded in is'D, T remains bounded in is'D' In cases such as those of Euclidean space where we have at our disposal an elementary kernel which we know possesses the required properties, this proof is certainly shorter than that of § 29. The advantage of the latter is that it precisely dispenses with the preliminary study of elementary kernels. We find moreover another proof of this same regularity theorem in Warner [1].

§31. The Equation !JS= T on a Compact Space. The Operators Hand G Given a current T on a Riemannian space V, if there exists a current S satisfying the equation ,1S = T, we have from (2, § 25), for each harmonic form ¢ with compact support, (T, ¢ )=(,1S, ¢) =(S, ,1¢)=O.

§31. The Equation LlS= T on a Compact Space. The Operators Hand G

131

F or the equation Ll S = T to be solvable, it is thus necessary that T be orthogonal to all harmonic forms with compact support. We will show that, when V is compact, this necessary condition is also sufficient l2 . A harmonic form ¢, satisfying Ll¢ =0, also satisfies QLl¢ = and wecan write this, by virtue of formula (I) of Lemma (3, § 28), as ¢ - Q' ¢ = 0. From the Fredholm theorem, by supposing the space V is compact, the solutions of this last equation form a finite dimensional vector space E. The harmonic forms thus constitute a subspace E' of E. Let E" be the complementary subspace orthogonal to E' in E. If we look for a solution of the equation Ll.u = [3, where [3 is a given form orthogonal to all harmonic forms, by putting .u=Q~, as in §30, we obtain the integral equation ~ - Q~ = [3, and from the Fredholm theory 13, this equation admits a solution if and only if [3 is othogonal to all forms satisfying the associated homogeneous equation ¢ - Q' ¢ = 0, that is, if [3 is orthogonal to E. However, [3 is only supposed to be orthogonal to E' so that the integral equation may not be able to be solved. Nevertheless, the differential equation Ll.u = [3 is able to be solved in the following way. If and bdGc/> are linear combinations with constant coefficients of second derivatives of the coefficients of Gc/>, to show that these forms are square summable, it is sufficient to show that, for each function c/> of ~, the second derivatives of Gc/> are square summable. This follows immediately from the fact that these second derivatives are O(e - n) as e~oo, where e denotes the distance of x from the origin. (We remark that, if n > 2, the first derivatives of G are square summable, and, if n > 4, Gc/> is square summable).

Chapter V. Harmonic Forms

150

It follows from this that, in Euclidean space as in a compact space, we can write HI

= dl5G and H2 = I5dG,

and this gives explicitly the kernels of HI and H2 and the forms occurring in formulas (3) and (7). Consider for example formula (3) in the plane. If the orientation of the plane is fixed, it suffices to consider even forms. We have, in rectangular coordinates, 1 n

1 r x,y

g(x'Y)=~2 log -(--) (1 +dx I . dyl +dx2 . dy2+dx I /\dx 2 . dyl /\dy2)

and

To obtain the terms of e(x,y) of degree 1 atx (and aty), it suffices to consider the term of degree zero in g(x,y). Denoting by 8(x,y) the angle formed by the vector joining x to y and the first coordinate axis, we have

and the set of terms of degree 1 in e(x,y) reduce to

If c and c' are two oriented plane curves such that the end points of each one are not on the other, formula (3) reduces to

It is clear that

S yec'

dy 8(x,y) is the angle with which we see c' from the pointx.1t is a

function of x, discontinuous on c', although its differential extends by continuity to points of c' which are not endpoints. The formula thus expresses that c /\ c' [1] is equal to the quotient by 2n of the difference between the continuous variation, along c, of the angle with which we see c', and the continuous variation, along c', of the angle with which we see c. This result is easy to verify directly. For n = 3, formula (7) reproduces in particular the classical formula of Gauss for the linking coefficient of two closed curves in the space.

§34. The Analyticity of Harmonic Forms

151

§ 34. The Analyticity of Harmonic Forms A Riemannian space V is said to be analytic if the manifold V is endowed with a real analytic structure and if the coefficients gij of di2, represented with the aid of local analytic coordinates, are real analytic functions of these coordinates. If this is the case, then in a neighbourhood of each point v of V, the normal coordinates with origin v are analytic; the coefficients of ds 2 represented with the aid of normal coordinates with origin v are thus real analytic functions of these coordinates in a neighbourhood of v. Conversely, if a Riemannian space Vis such that, in a neighbourhood of each point v, the coefficients of ds 2 represented with the aid of normal coordinates with origin v are analytic functions of these coordinates, there exists on Va well defined real analytic structure in which the real analytic functions at a point v are analytic functions of normal coordinates with origin v. We see that the analytic structure of an analytic Riemannian space is completely determined by its metric. On an analytic space, a differential form is said to be analytic if the coefficients of this form, represented with the aid of local analytic coordinates, are analytic functions of these coordinates. We will show that, on an analytic Riemannian space, the harmonic forms are analytic. More generally: Theorem 26. On an analytic Riemannian space, a current T such that Ll Tis equal to an analytic form in a domain D, is itself equal in D to an analytic form.

To establish this theorem 18 , it suffices to prove that, if Ll T is analytic in a neighbourhood of a point, T is also analytic in a neighbourhood of the same point. Let v be a point of Vand Dl a geodesic sphere with centre v and radius Rl which is sufficiently small so that each point of Dl is the end point of a unique geodesic arc in Dl from the origin v. With aid of normal coordinates Xl, . .. ,x" with origin v, orthonormal at v, Dl is defined by I (xif < Rf. We can choose Rl sufficiently small so that the function O"(x,y) which occurs as a factor in the expression (5, § 28) for the parametrix OJ (x, y) is equal to 1 whenever x and yare in D 1 • Moreover, we denote by D and Do the geodesic spheres with the same centre v and with radii satisfying 0 < R < Ro < Rl . It suffices to prove that, if Ll T is analytic in D 1 , T is analytic in D, where the radius R is a positive arbitrarily small number. Let Q be a Coo function with support in Dl and equal to 1 in Do. From Corollary 1 of Theorem (21, § 29), if Ll T is analytic in D 1 , T is Coo in D J and fl = Q T is a form with support in Dl which is everywhere ex> and which equals T in Do, so that Llfl is analytic in Do. We thus reduce to having to prove the following proposition: If fl is a Coo form with compact support in Dl such thqt Ll fl is analytic in Do, then fl is analytic in D. 18 The analytic character of harmonic forms has been established by Kodaira [2] using a method of Hadamard. Theorem 26 follows from it with the aid of the Cauchy-Kowalewski theorem (see Bochner [2]). The idea of the method followed here is due to E. E. Levi [2].

Chapter V. Harmonic Forms

152

The proof will be based on formula I of Lemma (3, § 28), QLJp, = p, - Q' p,. As the supports of It and LJp, are contained in D 1 , we can restrict to DI the domain of integration of the integrals defining QLJp, and Q'p,. By decomposing D1 into D and DI ~ D, the formula can be written as fleX) -

S

q(y, x)

1\

*yp,(y) = l/J(x),

YED

with

YED,-D

YED,

Further, denoting, for brevity, by P the operator defined by (2)

P4>(x)=

S

q(y, x)

1\

*yp,(y),

YED

our formula may be written finally as (3)

By the Liouville-Neumann iteration method used in § 30, we can extract p, as a function of l/J. Denote by 14>1 the least upper bound of the modulus of the coefficients of the form 4>, represented with the aid of the normal coordinates Xl, . .. , x n , in D. We have an inequality of the form

where k is a positive number independent of 4> which tends to zero with the radius R of D. Thus, IpmtjJ I~ kmll/J I, and noting the identity

it follows from (3) that, if R is sufficiently small so that k < 1, the form p, is equal to the sum of the uniformly convergent series OC!

(4)

fl= L

pn'l/J.

m=O

Consider then the space IR 2n of points x = (xl, ... ,xn) with complex coordinates. We can consider this as a 2n dimensional Euclidean space in which the Euclidean distance d(x, y) of two points x and y is defined by d2(x,Y)=L Ix i -y i I2. i

§34. The Analyticity of Harmonic Forms

153

Let IRn and J" be subspaces of IR2 n formed respectively by points with real and with purely imaginary coordinates. They intersect orthogonally at the origin which represents the point v of our space. Let D, Do and Dl be the interiors of spheres with centre v (that is, at the origin) in IR 2 n and radii R, Ro and Rl respectively. The intersections of these spheres with IR n represent D, Do and Dl respectively. A function is said to be holomorphic in a domain ofIR2n ifit is a holomorphic function of Xl, ... , xn in this domain. A differential form, represented in such a domain by the expression

will be said to be holomorphic if the coefficients (h ... i p are holomorphic. If these coefficients are bounded, we say that the form is bounded. Denote by x' and x" the orthogonal projections of a point x of IR2n onto IR n and J" respectively; we can call x' the real part of x and x" its imaginary part, and we can write x = x' + x". Let D Y be the set of points x of IR 2n such that x' is in D and

d(x, x') < y(R -d(x, x")). The radius (straight line segment in IR2n) joining a point x of DY to some point on the frontier B of D in IR n forms an angle with IR n which has sine < y. Thus, we will show that, if the positive numbers Rand yare sufficiently small, we have the two following propositions. A. If rfJ is a holomorphic form bounded in DY, PrfJ can be extended to a holomorphic form bounded in D Y, and, denoting by IrfJ Iy the least upper bound of the modulus of the coefficients of rfJ in DY, we have an inequality of the form

where ky is a positive number independent of rfJ which tends to zero with R. B. The form 1/1, defined in D by (1), extends to a holomorphic form bounded in DY. First, let us show that the proposition we are considering follows immediately from these two assertions. They imply in fact that the form prnl/l extends to a holomorphic form bounded in DY satisfying

where we can suppose ky < 1. Formula (4) then shows that fl is the sum of a uniformly convergent series in D Y of hoi omorphic forms in D Y and this ensures us that it is hoi om orphic in DY and thus analytic in D. It thus remains only to establish A and B. For this, we remark that the coefficients gij of ds 2 which are analytic functions in Dl extend to holomorphic functions in a domain of IR2 n containing D l . By

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154

choosing Rl sufficiently small, we can suppose that it is holomorphic in- jjl' The Christoffel symbols are thus also holomorphic in jjl, and by virtue of the classical existence theorems applied to the differential equations for the geodesics (1, § 27), there exists a unique natural motionz(t) =z(t; x, 0 satisfying the initial conditions z(O) = x and i(O) = ~ for any point x of Dl and vector ~ tangent to x. The point z (that is, each of its coordinates) is a holomorphic function of t, x and ~ provided that It Idoes not exceed a positive number which is a continuous function ofx and~. We call the curve described by z(t) when tvaries through real values a geodesic line. Through each point x of jjl passes a unique geodesic line tangent to a given vector ~ at x. The square of the Riemannian length of a vector ~ tangent to a point x of jjl' defined for complex x or ~ by

nl

is generally a complex number. It can vanish without ~ being zero: the vectors of zero Riemannian length form at each point a 2n - 2 dimensional cone. If the tangent vector of a point of a geodesic line has zero Riemannian length, so too do the tangent vectors at all other points of this geodesic line and this is then said to be a geodesic line of zero Riemannian length. The set of these lines passing through a given point forms moreover a (2n - 2) dimensional cone. The same argument as that used in § 27 shows that there exists a positive function Q(x), depending continuously on x, such that each point of the Euclidean sphere with centre x and radius g(x) can bejoined to x by a unique arc of a geodesic line contained in this sphere. On the other hand, the Euclidean radius of curvature of geodesic lines, in a neighbourhood of the origin v, has a positive greatest lower bound. We are allowed to suppose that Rl is smaller than this greatest lower bound and smaller than the greatest lower bound of h2(x) in D 1 . If then x and yare two points of jjl, as their Euclidean distance is less than 2Rl which is < Q(x), there is an arc of a geodesic line joining them which is contained in the Euclidean sphere with centre x and radius Q(x); this arc is necessarily contained in jjl since if not, by moving y towards x in jjl' it would become at one instant tangent and interior to the surface of the sphere bounding jjl' But this is impossible as its radius of curvature at the point of contact would have to be greater than R 1 • Thus, provided that Rl is sufficiently small, there is a unique arc of a geodesic line in jjl joining any two given points of jjl . The initial velocity ~ which must be assigned at a point so that in a natural motion, during the time interval 0;£ t;£ 1, the arc of the geodesic line joining x to y in jjl is described, is a holomorphic function of x and y in jjl' This is also the case for the square of its Riemannian length, which we will again call the square of the Riemannian distance of x to y,

This function vanishes only when the geodesic line joining x to y has zero Riemannian length. If ~ tends to 0, y tends to x and conversely. Moreover, the

§34. The Analyticity of Harmonic Forms

155

difference y - x - ¢ between the vector from the origin x to the end point y and the vector ¢ is infinitely small to the second order with respect to d(x, y). By choosing R sufficiently small, we can thus obtain thatfor any two points x and y of D, the angle between the vectors y -x and ¢ is less than a given arbitrarily small positive number. On the other hand, at each point of D, the angle of a vector of zero Riemannian length with JR.n is greater than a fixed positive number (this follows from the positive definite nature of ds 2 in the real domain). It is the same at each point of DY, provided y sufficiently small. It follows from this and the preceding that, provided y is sufficiently small, if x and yare two points of D Y such that the angle of the vector y -x with JR.n is less than arcsin y, the angle of ¢ with each vector (having the same origin x) of zero Riemannian length is greater than a fixed positive number. This implies that, whenever x and y vary so that this condition is satisfied, the ratio of the Euclidean length of ¢ and the modulus Ir(x,y)1 of its Riemannian length lies between two fixed positive numbers, and this is also true for the ratio Ir(x,y)l/d(x,y). In the complex domain, r(x,y) is a many-valued function of two signs. For pairs (x,y) satisfying the above condition, we choose one of the signs in the following way. We simultaneously move x and y with constant velocity along the normals to JR." until they meet their projections x' and y'. The angle of y - x with JR.n decreases to zero, and thus is always less than arcsin y and r2(x,y) can never vanish. Each sign of rex, y) is thus uniform along the path described by (x, y) and we will choose that one which is positive for x=x' and y=y'. The sign thus chosen is a holomorphic function ofx andy on the set of pairs (x, y) satisfying the above condition. Having made these remarks, to establish A, we observe that each coefficient of Pcp is the sum of a finite number of terms such that (5)

F(x)=

J p(x,y)f(y)dyl ... dyn YED

where p(x,y) is a coefficient of q(x,y) and fey) is a coefficient of *cp(y). By hypothesis,f(y) extends to a holomorphic function bounded in DY, and it suffices to show that F(x) also extends to a holomorphic function bounded in DY and that we have an inequality of the form IF(x)ly ~ lylf(x)ly between the upper bounds If(x)ly and IF(x)I). of the modulus of these functions in Dy where ly is a positive number independent of f(x) which tends to zero with the radius R of D. The proof of Lemma (1, § 28) shows that, in the actual case when r2(x, y) is a holomorphic function of x and y, each coefficientp(x,y) of q(x, y) is of the form p(x,y) = r-n(x,Y)PI (x, y),

where PI (x, y) is a holomorphic function of x and y in D I , which vanishes in the same way as its first derivatives for X= y, and this implies that PI (x, y) = O(d2(x,y)). It follows that we have (6)

p(x, y) = O(d 2 - n(x,y))

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156

whenever x and y vary in D in such a way that they do not coincide and so that the angle of the vector y - x with IR" remains less than arcsin y. Let then D(x) be the n dimensional cone in 1R2" with vertex x and which is bounded by the frontier B of D in IR". This cone gives rise to the radii joining x to points of B, and we put, for xEDY, (7)

J

F(x) =

p(x, y)f(y)dyl

1\ ••• 1\

dy".

yeD(x)

Evidently, for XED, expression (7) reduces to (5), as then D(x)=D. We show that the integral (7) is absolutely and uniformly convergent with respect to x for xEDY, and bounded. We take as the variables of integration the coordinates y'l, . .. ,yin of the orthogonal projection y' of yin IR" (the real part of y); y describes D(x) when y' describes D, and its coordinates yi are functions of y'i for which the Jacobian remains bounded. Since, on the other hand, relation (6) is valid for YED(x) and since the ratio of d(x,y) with d(x',y') thus lies between two fixed positive numbers, the integral (7) is transformed into F(x)=

J

d 2 -"(x',y')P2 (x,y)f(y)dy'l ... dyln

y'eD

where P2 (x, y') is a function which remains bounded. It follows immediately that the integral is absolutely and uniformly convergent and that between the least upper bounds IF(x)ly and If(x)ly of the modulus of the functions F(x) andf(x) in D we have the relation where Iy =

J

d 2 - "(x', y')lp2 (x, y')1 dy'l ... dyln

y'eD

tends to zero with the radius R in D. To show moreover that the function F(x) defined by (7) is holomorphic in D Y, we will put, by denoting by Ya variable point on the frontier B of D, y=x+t(Y -x).

When Y describes Band t varies from 0 to 1, y describes the cone D(x). Denoting by 1] the (n -1) dimensional area element of B, expression (7) transforms to 1

F(x)=J dt o

J P3(X, Y,t)f(x+t(Y-x»1],

yeB

where the function under the integral sign, P3(X, Y, t)f(x+ t(Y -x», is holomorphic with respect to x for t> O. The function 1

J J P3(X, Y, t)f(x+t(Y -x»1]

F,(x) = dt ,

yeB

157

§34. The Analyticity of Harmonic Forms

is then a holomorphic function of x in DY, and since it converges uniformly to F(x) as E~O, F(x) is also holomorphic in DY. Thus, proposition A is established. To establish proposition B, we partition expression (1) for ljJ(x). By putting J/1={3, we can write

+S

w(x,y)J\*y{3(y).

YED

For YED! ~ D and xEDY, the coefficients of the forms occurring under the integral sign in the first two terms of the second member are holomorphic functions of x and are O(d2 - n (x,y)). This immediately implies that these first two terms are holomorphic forms bounded in DY. Finally, as {3 = J/1 is holomorphic and bounded in DY, by virtue of the hypothesis, then, provided that }' is sufficiently small, an identical reasoning to that used to establish proposition A shows that the third term extends to a holomorphic form bounded in DY. This completes the proof. D The preceding method applies to other equations. Consider for example (8)

Using formula I of Lemma (3, § 28), we see that each solution /1 of (8) satisfies in D an integral equation of the form

/1 (x) -

J {q(y, x) -cw(x, y)} J\ *y/1(Y) = X(x) YED

which we can write as /1- Pc = X. The operator Pc possesses moreover the properties stated for P in proposition A, for any constant or analytic function c. If {3 is also analytic, we verify as above that Xextends to a holomorphic function bounded in DY. Thus: On an analytic Riemannian space, each solution /1 ofequation ( 8 ) is analytic in each domain where the form {3 and the function c are analytic. The fact that harmonic forms are analytic implies the following consequences. Corollary 1. A harmonic form on a connected, analytic Riemannian space such that at a point each derivative of each of its coefficients vanishes (this includes derivatives of order zero) is identically zero. Corollary 2. On a connected, analytic Riemannian space V, the support of a harmonic form which is not identically zero is the entire space.

Are these propositions, which apply also to the solutions of the equation J/1 + C/1 = 0 for any constant c, moreover valid in the case of a Coo Riemannian space which is not however analytic? The answer to this question, asked in the

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158

first edition of this book, is that they are. N. Aronszjan [1], Krzywicki and Szarski have in fact established a theorem which, even with very much weaker differentiability hypotheses, shows that a form u satisfying du = bu = 0 vanishes identically provided that it has at a point a zero of infinite order. For a form u which is Coo in a Coo Riemannian space, a zero of infinite order is a point where all the coefficients of u and all their derivatives are zero. Thus, it is also a zero of infinite order of du, bu, dbu and bdu. If u is harmonic and has a zero of infinite order, the form v = dbu = -bdu satisfying dv = bv = 0 is identically zero. It is the same for the form w = bu as dw = bw = 0 and likewise for duo This implies that u is also zero. Thus: a harmonic form which has a zero of infinite order is identically zero.

§ 35. Square Summable Harmonic Forms on a Complete Riemannian Space We will establish here the following theorem, due to A. Andreotti [1] and E. Vesentini (see also, Garland [1]).

Theorem 26. On a complete, non-compact Riemannian space V, each square summable harmonic form is closed and co closed. We commence with two remarks. If oc and [3 are two forms of the same degree, we put Q(oc, [3) =*(oc /\ *[3) and Q(oc) = Q(oc, oc) so that (oc, [3) = JQ(oc, [3h1.

We can call Q(oc) the local scalar square of oc; it is equal to the sum of the squares of the coefficients of the form oc relative to a coordinate system orthonormal at the point considered. If u is another form of degree q, the coefficients of the form O!t /\ oc, of degree p + q, are the sums of products, modified by a ± sign, of a coefficient of u by a coefficient of oc, and each of these products occurs at most once in the expression for the coefficients of u /\ OC. It follows immediately from the Cauchy-Schwarz inequality that we have (1)

Q(u /\ oc) ~ Q(O!t) Q(oc).

As V is complete, the set of points x of V for which distance r(x)=r(x,O) from a fixed point 0 is ~ R is compact for all R> O. The function rex) is continuous and, by the triangle inequality, satisfies Ir(y) -r(x)1 ~ rex, y). This function is generally not everywhere differentiable, but by regularising it by the method of § 15, we obtain a slightly different function Q(x) which is Coo such that the set BR={xIQ(x)~R} is compact for all R>O and satisfies IQ(Y)-Q(x)1 ~Mr(x,y) with a constant M> 1. Since Q(dQ) is simply the square of the gradient of Q, it follows that Q(dQ)~M2. Let now f.1.(t) be a Coo function of the real variable t such that O~f.1.(t)~l, f.1.(t) = 1 if t~l and f.1.(t) =0 if t~2. The

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159

function ),(x) =.u C2~)) possesses the following properties: O;:::;),(x);:::; 1, A(X)=O if

(2)

xEV~B2R'

A(x)=l if xEBR

Q(dA);:::;; (K=constant, independent of R).

Consider then a square summable harmonic form q;: L1q; =0 and (q;, q;) < As the form a =}, 2q; has compact support, we have

00.

(dq;, da) = (l5dq;, a).

On the other hand, drx = A2 dq; + 2 AdA A q; and (dq;, drx) = (dq;, A2dq;) + (dq;, 2 AdA A q;)

and thus, we deduce, using the identity (f, Ag) = (Af, g), (Adq;, Adq;) = (15 dq; , a) -(Adq;, 2dA

(3)

A

q;).

Analogously, (15q;, l5a) = (dl5q;, rx) and

imply (4)

As L1q;=0, (3) and (4) imply

Using the inequality I(f, g)1 ;:::;t(f,f) +t(g, g), which follows from the Schwarz inequality we obtain the following bounds for the absolute value of terms of the second member of (5):

q;)1 ;:::;t(Adq;, ),dq;) + 2(dA A q;, dA A q;), IAI5q;, *(2d), A *q;)1 ;:::;t(},I5q;, ),I5q;) + 2(dJe A *q;, d}, A *q;). I(Adq;, 2dA A

Thus, (5) implies: (6)

(Adq;, Adq;)+(AI5q;, JeI5q;);:::;4(dA A q;, dJe

A

q;)+4(dA A *q;,dA

A

*q;).

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160

From (1) and (2), we have on the other hand Q(dl/\