Integral Geometry and Valuations 3034808739, 978-3-0348-0873-6, 978-3-0348-0874-3

Table of contents :
Front Matter....Pages i-viii
New Structures on Valuations and Applications....Pages 1-45
Algebraic Integral Geometry....Pages 47-112
Back Matter....Pages 113-113

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Advanced Courses in Mathematics CRM Barcelona

Semyon Alesker Joseph H.G. Fu

Integral Geometry and Valuations

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta

More information about this series at http://www.springer.com/series/5038

Semyon Alesker • Joseph H.G. Fu

Integral Geometry and Valuations Editors for this volume: Eduardo Gallego, Universitat Autònoma de Barcelona Gil Solanes, Universitat Autònoma de Barcelona

Semyon Alesker Department of Mathematics Tel Aviv University Tel Aviv, Israel

Joseph H.G. Fu Department of Mathematics University of Georgia Athens, GA, USA

ISSN 2297-0304 ISSN 2297-0312 (electronic) ISBN 978-3-0348-0873-6 ISBN 978-3-0348-0874-3 (eBook) DOI 10.1007/978-3-0348-0874-3 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2014952675

Mathematics Subject Classification (2010): Primary: 52B45, 53C65; Secondary: 52A39 © Springer Basel 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

Foreword This book contains a revised and expanded version of the notes of the lectures given by Semyon Alesker and Joseph H. G. Fu in the Advanced Course on Integral Geometry and Valuation Theory that took place from September 6th to 10th, 2010 at the Centre de Recerca Matem`atica (CRM) in Bellaterra, Barcelona. This activity was intended as a modern introduction to integral geometry, with a special emphasis on the new ideas coming from the theory of convex valuations. Valuations are finitely additive functionals on the space of convex bodies. Hadwiger’s famous theorem characterizes continuous, rigid-motion-invariant valuations as linear combinations of the intrinsic volumes. This deep result allowed an axiomatic approach to integral geometry, although only in Euclidean space. After Hadwiger, the study of valuations has been an active subject, with essential contributions by P. McMullen, D. Klain, R. Schneider and others. A breakthrough in the theory happened in 2001, when S. Alesker proved his irreducibility theorem. This is a strong result that led to the discovery of several natural structures on the space of valuations. Among them, the algebra structure on the space of valuations is probably the most useful. Indeed, this structure is in a sense dual to the kinematic formulas that lie at the base of integral geometry. As shown by A. Bernig and J. Fu, the study of this algebraic structure of the valuation space provides a way to successfully determine the integral geometry of many spaces that were out of reach with the classical methods. The Advanced Course on Integral Geometry and Valuation Theory reported on this recent progress, providing at the same time an introduction to the subject. There were two series of lectures, delivered by Semyon Alesker and Joseph Fu. These lectures were complemented with several invited and contributed talks by ´ J. C. Alvarez-Paiva, A. Bernig, L. M. Cruz-Orive, N. Dutertre, F. Fodor, D. Hug, T. Leinster, E. Vedel-Jensen, and S. Willerton. We would like to express our gratitude to the director and the staff of the Centre de Recerca Matem` atica for making possible this activity. We also thank the Ministerio de Ciencia e Innovaci´ on of the Spanish government (refs. MTM20090876-E and MTM2009-06054-E) and the Consolider Ingenio Mathematica programme (ref. MIGS-C5-0328) for providing financial support. Special thanks are due to S. Alesker and J. Fu for the enthusiasm they showed towards this course and the careful preparation of these notes.

Eduardo Gallego and Gil Solanes

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Contents

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New Structures on Valuations and Applications Semyon Alesker

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Translation-invariant valuations on convex sets . . . . . . . . . . . 1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 McMullen’s theorem and mixed volumes . . . . . . . . . . . 1.1.3 Hadwiger’s theorem . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Irreducibility theorem . . . . . . . . . . . . . . . . . . . . . 1.1.5 Klain–Schneider characterization of simple valuations . . . 1.1.6 Smooth translation-invariant valuations . . . . . . . . . . . 1.1.7 Product on smooth translation-invariant valuations and Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Pull-back and push-forward of translation-invariant valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.10 Hard Lefschetz type theorems . . . . . . . . . . . . . . . . . 1.1.11 A Fourier-type transform on translation-invariant convex valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.12 General constructions of translation-invariant convex valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.13 Valuations invariant under a group . . . . . . . . . . . . . . 1.2 Valuations on manifolds . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definition of smooth valuations on manifolds and basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Canonical filtration on smooth valuations . . . . . . . . . . 1.2.3 Integration functional . . . . . . . . . . . . . . . . . . . . . 1.2.4 Product operation on smooth valuations on manifolds and Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Generalized valuations and constructible functions . . . . . 1.2.6 Euler–Verdier involution . . . . . . . . . . . . . . . . . . . . 1.2.7 Partial product operation on generalized valuations . . . . . 1.2.8 A heuristic remark . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 A few examples of computation of the product in integral geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.10 Functorial properties of valuations . . . . . . . . . . . . . . 1.2.11 Radon transform on valuations on manifolds . . . . . . . . 1.2.12 Khovanskii–Pukhlikov-type inversion formula for the Radon transform on valuations on RPn . . . . . . . . . . . . . . .

1 3 3 4 5 6 7 8 8 11 13 14 14 18 21 22 22 25 25 26 27 29 30 31 31 33 36 38

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Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Algebraic Integral Geometry Joseph H. G. Fu Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . 2.1 Classical integral geometry . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Intrinsic volumes and Federer curvature measures . . . . . . 2.1.2 Other incarnations of the normal cycle . . . . . . . . . . . . 2.1.3 Crofton formulas . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The classical kinematic formulas . . . . . . . . . . . . . . . 2.1.5 The Weyl principle . . . . . . . . . . . . . . . . . . . . . . . 2.2 Curvature measures and the normal cycle . . . . . . . . . . . . . . 2.2.1 Properties of the normal cycle . . . . . . . . . . . . . . . . 2.2.2 General curvature measures . . . . . . . . . . . . . . . . . . 2.2.3 Kinematic formulas for invariant curvature measures . . . . 2.2.4 The transfer principle . . . . . . . . . . . . . . . . . . . . . 2.3 Integral geometry of Euclidean spaces via Alesker theory . . . . . . 2.3.1 Survey of valuations on finite-dimensional real vector spaces 2.3.2 Constant coefficient valuations . . . . . . . . . . . . . . . . 2.3.3 The FTAIG for isotropic structures on Euclidean spaces . . 2.3.4 The classical integral geometry of Rn . . . . . . . . . . . . . 2.4 Valuations and integral geometry on isotropic manifolds . . . . . . 2.4.1 Brief definition of valuations on manifolds . . . . . . . . . . 2.4.2 First variation, the Rumin operator, and the kernel theorem 2.4.3 The filtration and the transfer principle for valuations . . . 2.4.4 The FTAIG for compact isotropic spaces . . . . . . . . . . 2.4.5 Analytic continuation . . . . . . . . . . . . . . . . . . . . . 2.4.6 Integral geometry of real space forms . . . . . . . . . . . . . 2.5 Hermitian integral geometry . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Algebra structure of ValU (n) (Cn ) . . . . . . . . . . . . . . . 2.5.2 Hermitian intrinsic volumes and special cones . . . . . . . . 2.5.3 Tasaki valuations and a mysterious duality . . . . . . . . . 2.5.4 Determination of the kinematic operator of (Cn , U (n)) . . . 2.5.5 Integral geometry of complex space forms . . . . . . . . . . 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 47 47 48 48 49 49 50 53 53 54 57 60 60 61 62 64 67 67 70 73 76 79 79 81 85 86 87 91 94 95 97 100 102 103 105 109

Chapter 1

New Structures on Valuations and Applications Semyon Alesker Introduction The theory of valuations on convex sets is a classical part of the topic of onvexity, with traditionally strong relations to integral geometry. During the roughly last 15 years a considerable progress was made in valuation theory and its applications to integral geometry. The progress is both conceptual and technical: several new structures on valuations have been discovered, new classification results of various special classes of valuations have been obtained, the tools used in valuation theory and its relations with other parts of mathematics have become much more diverse —besides convexity and integral geometry, one can mention representation theory, geometric measure theory, elements of contact geometry, and complex and quaternionic analysis. This progress in valuation theory has led to new developments in integral geometry, particularly in Hermitian spaces. Some of the new structures turned out to encode in an elegant and useful way important integral geometric information: for example, the product operation on valuations encodes somehow the principal kinematic formulas in various spaces. Quite recently, generalizations of the classical theory of valuations on convex sets to the context of manifolds were initiated; this development extends the applicability of valuation theory beyond affine spaces, and also covers a broader scope of integral geometric problems. In particular, the theory of valuations on manifolds provides a common point of view on three classical and previously unrelated directions of integral geometry: Crofton-style integral geometry, dealing with integral geometric and differential geometric invariants of sets and their intersections, and with projections to lower-dimensional subspaces; Gelfand-style integral geometry,

© Springer Basel 2014 S. Alesker, J.H.G. Fu, Integral Geometry and Valuations, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0874-3_1

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Chapter 1. New Structures on Valuations and Applications

dealing with the Radon transform on smooth functions on various spaces; and, less classical but still well known, the Radon transform with respect to the Euler characteristic on constructible functions. Although the relations between valuation theory and Crofton-style integral geometry have been known since the works of Blaschke and especially Hadwiger, the new developments have enriched both subjects and, in fact, more progress is expected. The relations of valuation theory to the two other types of integral geometry are new. Besides new notions, theorems, and applications, these recent developments contain a fair amount of new intuition on the subject. However, when one tries to make this intuition formally precise, the clarity of basic ideas is often lost among numerous technical details; moreover, in a few cases this formalization has not been done yet. Here, in several places, I take the opportunity to use the somewhat informal format of lecture notes to explain the new intuition in a heuristic way, leaving the technicalities aside. Nevertheless, I clearly separate formal rigorous statements from such heuristic discussions. The goal of my and Joe Fu’s lectures is to provide an introduction to these modern developments. These two sets of lectures complement each other. My lectures concentrate mostly on valuation theory itself and provide a general background for Fu’s lectures. In my lectures the discussion of the relations between valuation theory and integral geometry is usually relatively brief, and its goal is to give simple illustrations of general notions. The important exceptions are Sections 1.2.11 and 1.2.12, where new integral geometric results are discussed, namely a Radon-type transform on valuations. A much more thorough discussion of applications to Crofton-style integral geometry, especially in Hermitian spaces, will be offered in Fu’s lectures. My lectures consist of two main parts. The first part discusses the theory of valuations on convex sets and the second part discusses its recent generalizations to manifolds. The theory of valuations on convex sets is a very classical and much studied area. In these lectures, I mention only several facts from these classical developments which are necessary for our purposes; I refer to the surveys [54, 55] for further details and history. The exposition contains almost no proofs. I tried to give the necessary definitions and list the main properties and sometimes present constructions of the principal objects and some intuition behind. Among important new operations on valuations are product, convolution, Fourier-type transform, pull-back, pushforward, and the Radon-type transform on valuations; all of them are relevant to integral geometry and are discussed in these notes. Several interesting recent developments in valuation theory are not discussed here. The main omissions are a series of investigations by M. Ludwig with collaborators of valuations with weaker assumptions on continuity and various symmetries (see, e.g., [48, 50, 51]) and convex-bodies-valued valuations (see, e.g., [47, 49, 60]). Particularly, let me mention the surprising Ludwig–Reitzner characterization [50] of the affine surface area as the only example (up to the Euler characteristic, vol-

1.1. Translation-invariant valuations on convex sets

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ume, and a non-negative multiplicative factor) of upper semi-continuous convex valuation invariant under all affine volume-preserving transformations.

Acknowledgements These are notes of my lectures given at Centre de Recerca Matem`atica during the Advanced Course on Integral Geometry and Valuation Theory; I thank this institution and the organizers of the course E. Gallego, X. Gual, G. Solanes, and E. Teufel, for the invitation to give these lectures. I thank A. Bernig for his remarks on the first version of the notes, and E. Gallego, F. Schuster, and G. Solanes for a very careful reading of them and numerous remarks.

1.1 Translation-invariant valuations on convex sets 1.1.1 Definitions Let V be a finite-dimensional vector space of dimension n. Throughout these notes we will denote by K(V ) the family of all convex compact non-empty subsets of V . Definition 1.1.1. A complex-valued functional φ : K(V ) −→ C is called a valuation if φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B) whenever A, B, A ∪ B ∈ K(V ). Remark 1.1.2. In Section 1.2 we will introduce a different but closely related notion of valuation on a smooth manifold. To avoid confusion, we will sometimes call valuations on convex sets from Definition 1.1.1 convex valuations, though this is not a traditional terminology. But when there is no danger of confusion, we will call them just valuations. In fact, all valuations from Section 1.1 will be convex, while those from Section 1.2 will not, unless otherwise stated. Examples 1.1.3. (1) Any C-valued measure on V is a convex valuation. In particular, the Lebesgue measure is such. (2) The Euler characteristic χ, defined by χ(K) = 1 for any K ∈ K(V ), is a convex valuation. (3) Let φ be a convex valuation. Let C ∈ K(V ) be fixed. Define ψ(K) := φ(K + C). Then ψ is a convex valuation. (Here K + C := {k + c | k ∈ K, c ∈ C} is the Minkowski sum.) Indeed, (A ∪ B) + C = (A + C) ∪ (B + C), and if A, B, A ∪ B ∈ K(V ), then (A ∩ B) + C = (A + C) ∩ (B + C).

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Chapter 1. New Structures on Valuations and Applications

Let us define a very important class of continuous convex valuations. Fix a Euclidean metric on V . The Hausdorff distance on K(V ) is defined by   distH (A, B) := inf ε > 0 | A ⊂ (B)ε , B ⊂ (A)ε , where (A)ε denotes the ε-neighborhood of A in the Euclidean metric. It is well known (see, e.g., [58]) that K(V ) equipped with distH is a metric locally compact space in which any closed bounded set is compact. Definition 1.1.4. A convex valuation φ : K(V ) → C is called continuous if φ is continuous in the Hausdorff metric. This notion of continuity of a valuation is readily seen to be independent of the choice of a Euclidean metric on V . Definition 1.1.5. A convex valuation φ : K(V ) → C is called translation-invariant if φ(K + x) = φ(K) for any K ∈ K(V ), x ∈ V . The linear space of translation-invariant continuous convex valuations will be denoted by Val(V ). If equipped with the topology of uniform convergence on compact subsets of K(V ), Val(V ) is a Fr´echet space. In fact it follows from McMullen’s decomposition (Corollary 1.1.7 below) that Val(V ) with this topology is a Banach space, with the norm given by ||φ|| := sup |φ(K)|, K⊂D

where D ⊂ V is the unit Euclidean ball for some auxiliary Euclidean metric.

1.1.2 McMullen’s theorem and mixed volumes The following result due to McMullen [52] is very important. Theorem 1.1.6. Let φ : K(V ) → C be a translation-invariant continuous convex valuation. Then for any convex compact sets A1 , . . . , As ∈ K(V ) the function f (λ1 , . . . , λs ) = φ(λ1 A1 + · · · + λs As ), defined for λ1 , . . . , λs ≥ 0, is a polynomial of degree at most n = dim V . The special case s = 1 is already non-trivial and important. It means that, for λ ≥ 0, φ(λK) = φ0 (K) + λφ1 (K) + · · · + λn φn (K). It is easy to see that the coefficients φ0 , φ1 , . . . , φn are also continuous translationinvariant convex valuations. Moreover, φi is homogeneous of degree i (or i-homogeneous for brevity). By definition, a valuation ψ is called i-homogeneous if for any K ∈ K(V ) and any λ ≥ 0 one has φ(λK) = λi φ(K).

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Let us denote by Vali (V ) the subspace in Val(V ) of i-homogeneous valuations. We immediately get the following corollary: Corollary 1.1.7 (McMullen’s decomposition). Val(V ) =

n 

Vali (V ).

i=0

Remark 1.1.8. Clearly Val0 (V ) is one-dimensional and is spanned by the Euler characteristic. Actually, Valn (V ) is also one-dimensional and is spanned by a Lebesgue measure; this fact is not obvious and was proved by Hadwiger [39]. Let us now recall the definition of (Minkowski’s) mixed volumes, which provide interesting examples of translation-invariant continuous convex valuations. Fix a Lebesgue measure on V , denoted vol. For any n-tuple of convex compact sets A1 , . . . , An consider the function f (λ1 , . . . , λn ) = vol(λ1 A1 + · · · + λn An ). This is a homogeneous polynomial in λi ≥ 0 of degree n. Of course, this fact follows from McMullen’s theorem (Theorem 1.1.6) and the n-homogeneity of the volume, though originally it was discovered much earlier by Minkowski, and in this particular case there is a simpler proof (see, e.g., [58]). Definition 1.1.9. The coefficient of the monomial λ1 · · · λn in f (λ1 , . . . , λn ) divided by n! is called the mixed volume of A1 , . . . , An and is denoted by V (A1 , . . . , An ). The normalization of the coefficient is chosen in such a way that vol(A) = V (A, . . . , A). Mixed volumes have a number of interesting properties; in particular they satisfy the Aleksandrov–Fenchel inequality [58]. The property relevant for us, however, is the valuation property. Fix 1 ≤ s ≤ n − 1 and an s-tuple of convex compact sets A1 , . . . , As . Define   (1.1.1) φ(K) = V K[n − s], A1 , . . . , As , where K[n−s] means that K is taken n−s times. Then φ is a translation-invariant continuous valuation. This easily follows from Example 1.1.3(3).

1.1.3 Hadwiger’s theorem One of the most famous and classical results of valuation theory is Hadwiger’s classification of isometry-invariant continuous convex valuations on the Euclidean space Rn . To formulate it, let us denote by Vi the ith intrinsic volume, which by definition is   Vi (K) = cn,i V K[i], D[n − i] , where cn,i is an explicitly written constant which is just a standard normalization (see [58]). In particular, V0 = χ is the Euler characteristic and Vn = vol is the Lebesgue measure normalized so that the volume of the unit cube is equal to 1. Clearly Vi ∈ Vali is an O(n)-invariant valuation.

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Chapter 1. New Structures on Valuations and Applications

Theorem 1.1.10 (Hadwiger’s classification [39]). Any SO(n)-invariant translationinvariant continuous convex valuation is a linear combination of V0 , V1 , . . . , Vn . (In particular, it is O(n)-invariant.) In 1995, Klain [43] obtained a simplified proof of this deep result as an easy consequence of his classification of simple even valuations discussed below in Section 1.1.5. Hadwiger’s theorem turned out to be very useful in integral geometry of the Euclidean space. This will be discussed in more detail in J. Fu’s lectures. We also refer to the book [45].

1.1.4

Irreducibility theorem

One of the basic questions in valuation theory is to describe valuations with given properties. Hadwiger’s theorem is one example of such a result of great importance. In recent years, a number of classification results of various classes of valuations have been obtained. The case of continuous translation-invariant valuations will be discussed in detail in these lectures and in the lectures by Fu. The question is whether it is possible or not to give a reasonable description of all translation-invariant continuous convex valuations. In 1980, P. McMullen [53] formulated a more precise conjecture which says that linear combinations of mixed volumes (as in (1.1.1)) are dense in Val. This conjecture was proved in the positive by the author [2] in a stronger form which later on turned out to be important in further developments and applications. To describe the result, let us make a few more remarks. We say that a valuation φ is even (respectively odd ) if φ(−K) = φ(K) (respectively φ(−K) = −φ(K)) for any K ∈ K(V ). The subspace of even (respectively odd) i-homogeneous valu− ations will be denoted by Val+ i (respectively Vali ). Clearly, − Vali = Val+ i ⊕ Vali .

(1.1.2)

Next observe that the group GL(V ) of all invertible linear transformations acts linearly on Val by (gφ)(K) = φ(g −1 K). − Theorem 1.1.11 (Irreducibility theorem [2]). For each i, the spaces Val+ i and Vali are irreducible representations of GL(V ), i.e., they do not have proper invariant closed subspaces. + Remark 1.1.12. By Remark 1.1.8, Val+ 0 = Val0 and Valn = Valn are one-dimen± sional. But for 1 ≤ i ≤ n − 1 the spaces Vali are infinite-dimensional. Valn−1 was explicitly described by McMullen [53]; we state his result in Section 1.1.5 below.

Theorem 1.1.11 immediately implies McMullen’s conjecture. Indeed, it is easy to see that the closure of the linear span of mixed volumes is a GL(V )-invariant subspace, and its intersection with any Val± i is non-zero. Hence, by the irreducibility theorem, any such intersection should be equal to the whole space Val± i .

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The irreducibility theorem will be used in these lectures several times. The proof of this result uses a number of deep results from valuation theory in combination with representation-theoretical techniques. A particularly important such result of high independent interest is the Klain–Schneider classification of simple translation-invariant continuous convex valuations; it is discussed in the next section.

1.1.5 Klain–Schneider characterization of simple valuations Definition 1.1.13. A convex valuation φ ∈ Val is called simple if φ(K) = 0 for any K ∈ K(V ) with dim K < n := dim V . Theorem 1.1.14. (i) (Klain [43]). Any simple even valuation from Val is proportional to the Lebesgue measure. (ii) (Schneider [59]). Any simple odd valuation from Val is (n − 1)-homogeneous. Clearly, any simple valuation is the sum of a simple even one and a simple odd one. Hence, in order to complete the description of simple valuations, it remains to classify simple (n − 1)-homogeneous valuations. Fortunately, McMullen [53] has previously described Valn−1 very explicitly. His result was used in Schneider’s proof, and it is worthwhile to state it explicitly as it is of independent interest. First let us recall the definition of the area measure Sn−1 (K, · ) of a convex compact set K. Though it is not strictly necessary, it is convenient and common to fix a Euclidean metric on V . After this choice, Sn−1 (K, · ) is a measure on the unit sphere S n−1 defined as follows. First let us assume that K is a polytope. For any (n − 1)-face F , let us denote by nF the unit outer normal to F . Then, by definition,  voln−1 (F )δnF , Sn−1 (K, · ) = F

where the sum runs over all (n − 1)-faces of K, and δnF denotes the delta measure supported at nF . Then the claim is that the area measure extends uniquely by weak continuity to the class of all convex compact sets: if KN → K in the Hausdorff metric, then Sn−1 (KN , · ) → Sn−1 (K, · ) weakly in the sense of measures (see [58, §4.2]). Theorem 1.1.15 (McMullen [53]). Let φ ∈ Valn−1 and n = dim V . Then there exists a continuous function f : S n−1 → C such that  f (x) dSn−1 (K, x). φ(K) = S n−1

Conversely, any expression of this form with f continuous is a valuation from Valn−1 . Moreover, two continuous functions f and g define the same valuation if and only if the difference f − g is the restriction of a linear functional on V to the unit sphere.

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Chapter 1. New Structures on Valuations and Applications Now we can state the classification of simple valuations.

Theorem 1.1.16 (Klain–Schneider). Simple translation-invariant continuous valuations are precisely of the form  f (x) dSn−1 (K, x), K −→ c · voln (K) + S n−1

→ C is an odd continuous function and c is a constant. Moreover, where f : S the constant c is defined uniquely, while f is defined up to a linear functional. n−1

1.1.6 Smooth translation-invariant valuations We are going to describe an important subclass of Val, that of smooth valuations. They form a dense subspace in Val and carry a number of extra structures (e.g., product, convolution, Fourier transform) which do not extend by continuity to the whole space Val of continuous valuations. Moreover, main examples relevant to integral geometry are in fact smooth valuations. Definition 1.1.17. A valuation φ ∈ Val(V ) is called smooth if the Banach space valued map GL(V ) → Val(V ) given by g → g(φ) is infinitely differentiable. By a very general and elementary representation-theoretical reasoning, the subset of smooth valuations, denoted by Valsm (V ), is a linear dense subspace of Val(V ) invariant under the natural action of GL(V ). Also, Valsm (V ) carries a linear topology which is stronger than that induced from Val(V ), and with respect to which it is a Fr´echet space. This is often called the G˚ arding topology, and we will tacitly assume that Valsm (V ) is equipped with it. Of course, Valsm also satisfies McMullen’s decomposition and the irreducibility theorem. Let us give some examples of smooth valuations and non-smooth valuations. Let A, A1 , . . . , As ∈ K(V ) be full-dimensional convex bodies with infinitely smooth strictly convex boundary. Then K → vol(K + A) is a smooth valuation, and consequently the mixed volumes K → V (K[n − s], A1 , . . . , As ) are also smooth valuations. A simple geometric example of a non-smooth continuous valuation is the volume of a projection to a subspace of V . For future applications to integral geometry, the following result will be important. Proposition 1.1.18 ([4]). Let G be a compact subgroup of the orthogonal group of a Euclidean space V . Assume that G acts transitively on the unit sphere of V . Then any G-invariant valuation from Val(V ) is smooth.

1.1.7 Product on smooth translation-invariant valuations and Poincar´e duality In this section we discuss the product operation on translation-invariant smooth valuations introduced in [4]. This structure turned out to be intimately related to integral geometric formulas discussed in detail in J. Fu’s lectures.

1.1. Translation-invariant valuations on convex sets

9

First we introduce the exterior product on valuations. Theorem 1.1.19 ([4]). Let V and W be finite-dimensional real vector spaces. There exists a continuous bilinear map, called exterior product, Valsm (V ) × Valsm (W ) −→ Val(V × W ) which is uniquely characterized by the following property: fix A ∈ K(V ) and B ∈ K(W ) such that both of them have smooth boundaries with positive curvature. Let volV and volW be Lebesgue measures on V and W , respectively. Let φ(K) = volV (K + A) and ψ(K) = volW (K + B). Then their exterior product, denoted by φ  ψ, is (φ  ψ)(K) = volV ×W (K + (A × B)) for any K ∈ K(V × W ), where volV ×W is the product measure of volV and volW . Notice that the uniqueness in this theorem follows immediately from the (proved) McMullen’s conjecture, since linear combinations of valuations of the form vol( · + A) are dense in valuations. Let us emphasize that the exterior product is defined on smooth valuations, but it takes values not in smooth but just continuous valuations. Usually the exterior product is not smooth. Let us give some examples. Examples 1.1.20. (1) Obviously, the exterior product of Lebesgue measures in the sense of valuations coincides obviously with their measure-theoretical product. (2) The exterior product of Euler characteristics is the Euler characteristic on V × W . This can be seen as follows. First it is easy to verify that the exterior product commutes with the natural action of the group GL(V ) × GL(W ). Hence χV  χW is invariant under this group, and in particular 0-homogeneous. Thus it must be proportional to the Euler characteristic. Explicit evaluation of this product on a point shows that the coefficient of proportionality must be equal to 1. This evaluation can be done by observing that dn 1 volV (K + εD), χV (K) = n! volV (D) dεn ε=0 where D is a Euclidean ball (or any convex compact set of non-zero volume) and n = dim V . (3) Let volV be a Lebesgue measure on V and χW be the Euler characteristic on W . Then the exterior product volV χW is the volume of the projection to V : (volV χW )(K) = volV (prV (K)) for any K ∈ K(V × W ), where prV : V ×W → V is the natural projection. Observe that this valuation is not smooth (in contrast to the first two examples.)

10

Chapter 1. New Structures on Valuations and Applications

The first non-trivial point in Theorem 1.1.19 is that the exterior product is well defined; the second one is continuity. We do not give here any proof. However, let us give an incomplete, but instructive, explanation of why the exterior product is well defined. There are of course many different ways to write a valuation as a linear combination of vol( · + A). Let us check that the exterior product of finite linear combinations of such expressions is independent of the particular choice of a linear combination. Since the situation is symmetric with respect to both

valuations, we may assume that φ( · ) = i ci · volV ( · + Ai ) and ψ( · ) = volW ( · + B). Then using Fubini’s theorem and the equality Ai × B = (Ai × 0) + (0 × B) we get  (φ  ψ)(K) = ci · volV ×W (K + (Ai × B)) i

=

 

 ci ·

volV



 (K + (0 × B)) ∩ (V × {w}) + Ai d volW (w)

w∈W

i

 φ (K + (0 × B)) ∩ (V × {w}) d volW (w).

= w∈W

Clearly the last expression is independent of the form of presentation of φ. Now let us define the product on Valsm . Let us denote by Δ : V −→ V × V the diagonal imbedding. The product of φ, ψ ∈ Valsm (V ) is defined by (φ · ψ)(K) := (φ  ψ)(Δ(K)). It turns out that the product of smooth valuations is again smooth. Theorem 1.1.21 ([4]). The product of smooth valuations Valsm (V ) × Valsm (V ) → arding topology), associative, commutative, and Valsm (V ) is continuous (in the G˚ distributive. Accordingly, Valsm (V ) becomes an algebra over C with unit, which is the Euler characteristic. Moreover, the product respects the degree of homogeneity: sm sm Valsm i · Valj ⊂ Vali+j .

Example 1.1.22. The product of intrinsic volumes Vi ·Vj with i+j ≤ n is a non-zero multiple of Vi+j : by the Hadwiger theorem it is clear that the product should be proportional to Vi+j ; the constant of proportionality can be computed explicitly. Let explain why the Euler characteristic is a unit in Valsm (V ). Let φ(K) = voln (K + A). Fix a convex body B of non-zero volume. Then χ(K) =

1 dn voln (K + εB). n! voln (B) dεn ε=0

1.1. Translation-invariant valuations on convex sets

11

Consequently, (χ · φ)(K) =

1 dn vol2n ((Δ(K) + (0 × A)) + εB × 0) . n! voln (B) dεn ε=0

(1.1.3)

It is well known (prove it, or see an equivalent formula (5.3.23) in [58]) that for a convex compact subset U in a k-dimensional linear subspace Ek of the Euclidean space RN and any convex compact set M ⊂ RN , one has 1 dk volN (M + εU ) = volN −k (pEk⊥ M ). k! volk (U ) dεk ε=0 This and (1.1.3) imply that (χ · φ)(K) = voln (p1 (Δ(K) + (0 × A))) = voln (K + A) = φ(K), where p1 : V × V → V is the natural projection to the first component. An interesting property of the product is a version of Poincar´e duality. Theorem 1.1.23 ([4]). For any i = 0, 1, . . . , n = dim V the bilinear map sm sm Valsm i (V ) × Valn−i (V ) −→ Valn (V )

is a perfect pairing, namely for any non-zero valuation φ ∈ Valsm i (V ) there exists (V ) such that φ · ψ =  0. ψ ∈ Valsm n−i This result follows easily from the irreducibility theorem (Theorem 1.1.11). Indeed, it suffices to prove the statement for valuations of fixed parity ε = ±1. (V ) is a GL(V )-invariant closed Then the kernel of the above pairing in Valε,sm i subspace. Hence it must be either zero or everything. But it cannot be everything, ε,sm (V ) = 0. But since then for any valuation ψ ∈ Valsm n−i (V ) one would have ψ ·Vali this is not the case, as can be easily proved by constructing an explicit example. (Say in the even case, the product of the intrinsic volumes Vi · Vn−i is a non-zero multiple of the Lebesgue measure.) Thus Valsm (V ) is a graded algebra that satisfies Poincar´e duality. In Section 1.1.10 we will also see that it satisfies two versions of the hard Lefschetz theorem.

1.1.8 Pull-back and push-forward of translation-invariant valuations In this section we describe the operations of pull-back and push-forward on translation-invariant valuations under linear maps. Let f : V → W be a linear map. We define [15] a continuous linear map, f ∗ : Val(W ) −→ Val(V ) called pull-back, as usual by (f ∗ φ)(K) = φ(f (K)). It is easy to see that f ∗ φ is indeed a continuous translation-invariant convex valuation. The following result is evident.

12

Chapter 1. New Structures on Valuations and Applications

Proposition 1.1.24. (i) f ∗ preserves the degree of homogeneity and the parity. (ii) (f ◦ g)∗ = g ∗ ◦ f ∗ . Notice that the product on valuations can be expressed via the exterior product and the pull-back as φ · ψ = Δ∗ (φ  ψ), where Δ is the diagonal imbedding. A somewhat less obvious operation is the push-forward f∗ , introduced in [15]. In some non-precise sense f∗ is dual to f ∗ . In these notes it will be used only to give an alternative description of the convolution on valuations in Section 1.1.9 and to clarify some properties of the Fourier-type transform on valuations in Section 1.1.11; the reader not interested in these subjects may skip the rest of this section. Canonically, the push-forward map acts not between spaces of valuations, but between their tensor product (twist) with an appropriate one-dimensional space of Lebesgue measures. To be more precise, let us denote by D(V ∗ ) the onedimensional space of (C-valued) Lebesgue measures on V ∗ . Then f∗ is a linear continuous map f∗ : Val(V ) ⊗ D(V ∗ ) −→ Val(W ) ⊗ D(W ∗ ). In order to define this map, we will split its construction into the cases of f being injective, surjective, and a general linear map. Case 1: Let f be injective. Thus we may assume that V ⊂ W . In order to simplify the notation, we choose a splitting W = V ⊕ L and we fix Lebesgue measures on V and L. Then on W we have the product measure. These choices induce isomorphisms D(V ∗ )  C and D(W ∗ )  C. We leave for the reader to check that the construction of f∗ is independent of these choices. Let φ ∈ Val(V ). Define  (f∗ φ)(K) = φ(K ∩ (l + V )) dl. l∈L

It is easy to see that f∗ : Val(V ) → Val(W ) is a continuous linear map. Case 2: Let f be surjective. Again it will be convenient to assume that f is just a projection to a subspace, and fix a splitting V = W ⊕ M . Again fix Lebesgue measures on W, M , and hence on V . Let us also fix a set S ∈ K(M ) of unit measure. Set m := dim M . Then define 1 dm φ(K + ε · S) (f∗ φ)(K) = . m m! dε ε=0 Recall that by McMullen’s theorem φ(K +ε·S) is a polynomial in ε ≥ 0. Moreover, its degree is at most m. Indeed, when K is fixed, this expression is a translationinvariant continuous valuation with respect to S ∈ K(M ). The coefficient of εm is

1.1. Translation-invariant valuations on convex sets

13

an m-homogeneous valuation with respect to S ⊂ M , and hence, by Hadwiger’s theorem (see Remark 1.1.8), it must be proportional to the Lebesgue measure on M with a constant depending on K. By our definition, this coefficient is exactly (f∗ φ)(K) —in particular it does not depend on S. In fact, it also does not depend on the choice of Lebesgue measures and the splitting. Case 3: Let f be a general linear map. Let us choose a factorization f = g ◦h, where h : V → Z is surjective and g : Z → W is injective. Then define f∗ := g∗ ◦h∗ . One can show that f∗ is independent of the choice of such a factorization. Proposition 1.1.25 ([15, Section 3.2]). (i) The map f∗ : Val(V ) ⊗ D(V ∗ ) → Val(W ) ⊗ D(W ∗ ) is a continuous linear operator. (ii) (f ◦ g)∗ = f∗ ◦ g∗ . (iii) f∗ (Vali (V ) ⊗ D(V ∗ )) ⊂ Vali−dim V +dim W (W ) ⊗ D(W ∗ ).

1.1.9

Convolution

In this section we describe another interesting operation on valuations: a convolution introduced by Bernig and Fu [24]. This is a continuous product on Valsm ⊗D(V ∗ ). To simplify the notation, let us fix a Lebesgue measure vol on V ; it induces a Lebesgue measure on V ∗ . With these identifications, convolution is going to be defined on Valsm (V ) (without the twist by D(V ∗ )). Theorem 1.1.26 ([24]). There exists a unique continuous bilinear map, called convolution, ∗ : Valsm (V ) × Valsm (V ) −→ Valsm (V ), such that vol( · + A) ∗ vol( · + B) = vol( · + A + B). This product makes Valsm (V ) a commutative associative algebra with unit element sm sm vol. Moreover, Valsm i ∗ Valj ⊂ Vali+j−n . The above result characterizes the convolution uniquely, and allows to compute it in some examples. We can give, however, one more description of it using the previously introduced operations. Namely, let a : V × V → V be the addition map, a(x, y) = x + y. Then, by [15, Proposition 3.3.2], one has φ ∗ ψ = a∗ (φ  ψ). The product and convolution will be transformed into one another in Section 1.1.11 by another useful operation, the Fourier-type transform.

14

Chapter 1. New Structures on Valuations and Applications

1.1.10 Hard Lefschetz type theorems The product and the convolution on valuations enjoy another non-trivial property, analogous to the hard Lefschetz theorem from algebraic geometry [36]. Let us fix a Euclidean metric on V . Consider the operator sm L : Valsm ∗ −→ Val∗+1

given by Lφ := φ · V1 , where V1 is the first intrinsic volume introduced in Section 1.1.3. Consider also another operator sm Λ : Valsm ∗ −→ Val∗−1 , d defined by (Λφ)(K) = dε φ(K + ε · D) ε=0 , where D is the unit ball (here we use again McMullen’s theorem that φ(K + ε · D) is a polynomial). Notice that up to a normalizing constant, the operator Λ is equal to the convolution with Vn−1 , as was observed by Bernig and Fu [24] (here is a hint: use Theorem 1.1.26 and the fact that Λ(voln ) is equal to a constant times Vn−1 ).

→ Valsm Theorem 1.1.27. (i) Let 0 ≤ i < 12 n. Then Ln−2i : Valsm i n−i is an isomorphism. → Valsm (ii) Let 12 n < i ≤ n. Then Λ2i−n : Valsm i n−i is an isomorphism. Several people have contributed to the proof of this theorem. First the author proved (i) and (ii) in the even case [3, 6], using previous joint work with Bernstein [17] and integral geometry on Grassmannians (Radon and cosine transforms). Then Bernig and Br¨ ocker [23] proved part (ii) in the odd case, using a very different method: the Laplacian acting on differential forms on the sphere bundle and some results from complex geometry (K¨ ahler identities). Next Bernig and Fu have shown [24] that, in the even case, the two versions of the hard Lefschetz theorem are in fact equivalent via the Fourier transform (which was at that time defined only for even valuations). Finally, the author extended in [15] the Fourier transform to odd valuations and derived version (i) of the hard Lefschetz theorem in the odd case from version (ii).

1.1.11 A Fourier-type transform on translation-invariant convex valuations A Fourier-type transform on translation-invariant smooth valuations is another useful operation. First it was introduced in [3] (under the different name of duality) for even valuations and was applied there to Hermitian integral geometry in order to construct a new basis in the space of U (n)-invariant valuations on Cn . In the odd case it was constructed in [15]. Some recent applications and non-trivial computations of the Fourier transform in Hermitian integral geometry are due to Bernig and Fu [25]. In this section we will describe the general properties of the Fourier transform and its relation to the product and convolution described above. We will present

1.1. Translation-invariant valuations on convex sets

15

a construction of the Fourier transform in the even case only. The construction in the odd case is more technical. Notice that the even case will suffice for a reader interested mostly in applications to integral geometry of affine spaces, since by a result of Bernig [20] any G-invariant valuation from Val must be even, provided G is a compact subgroup of the orthogonal group acting transitively on the unit sphere. The main general properties of the Fourier transform are summarized in the following theorem. Part (2) for even valuations was proved in [24], while the general case and parts (1) and (3) were proved in [15]. Theorem 1.1.28. There exists an isomorphism of linear topological spaces F : Valsm (V ) −→ Valsm (V ∗ ) ⊗ D(V ) which satisfies the following properties: (1) F commutes with the natural actions of the group GL(V ) on the two spaces. (2) F is an isomorphism of algebras when the source is equipped with the product and the target with the convolution. (3) A Plancherel-type inversion formula holds for F, as explained below. In order to describe the Plancherel-type formula and present a more explicit description of the Fourier transform, it is convenient (but not necessary) to fix a Euclidean metric on V . This induces identifications V ∗  V and D(V ∗ )  C. With these identifications, F : Valsm (V ) → Valsm (V ); actually it changes the degree of homogeneity: ˜ Valsm F : Valsm i −→ n−i . The Plancherel-type formula says, under these identifications, that (F2 φ)(K) = φ(−K). Here are a few simple examples: F(vol) = χ; F(χ) = vol; F(Vi ) = cn,i Vn−i , where cn,i > 0 is a normalizing constant that can be computed explicitly. (Notice that the last fact, except for the positivity of cn,i , is an immediate consequence of the fact that F commutes with the action of O(n) and Hadwiger’s theorem.) The Fourier transform on a 2-dimensional space has an explicit description which we are going to give now. We will work for simplicity in R2 with the standard Euclidean metric and standard orientation. With the identifications induced by the metric as above, F : Valsm (R2 ) → Valsm (R2 ). It remains to describe F on 1homogeneous valuations. Every such smooth valuation φ can be written uniquely in the form  h(ω) dS1 (K, ω), φ(K) = S1

where h : S 1 → C is a smooth function which is orthogonal on S 1 to the 2-dimensional space of linear functionals. Let us decompose h into the even and odd parts: h = h+ + h− .

16

Chapter 1. New Structures on Valuations and Applications

Let us decompose further the odd part h− into “holomorphic” and “anti-holomorphic” parts, anti h− = hhol − + h− , as follows. Expand h− in the usual Fourier series on the circle S 1 : h− (ω) =



ˆ − (k)eikω . h

k

By definition, hhol − (ω) :=



ˆ − (k)eikω , h

k>0

hanti − (ω)

:=



ˆ − (k)eikω . h

k0 , where 0 is the zero-section of T ∗ X, and R>0 is the multiplicative group of positive real numbers acting on T ∗ X by multiplication on the cotangent vectors. We call PX the cosphere bundle since if one fixes a Riemann2 All manifolds are assumed to be countable at infinity, i.e., presentable as a union of countably many compact subsets.

1.2. Valuations on manifolds

23

ian metric on X, then it induces an identification of PX with the unit (co)tangent bundle. Let P ∈ P(X), and let x ∈ P be a point. The tangent cone to P at x is the subset of the tangent space Tx X consisting of all ξ such that there exists a C 1 -smooth curve γ : [0, 1] → P such that γ(0) = x and γ (0) = ξ. It it not hard to see that Tx P ⊂ Tx X is a closed convex polyhedral cone. Let (Tx P )o denote the dual cone, namely   (Tx P )o := η ∈ Tx∗ X | η, ξ ≤ 0 for any ξ ∈ Tx P . This is a closed convex cone in T ∗ X. Now define the normal cycle of P by

   N (P ) := (Tx P )o \{0} /R>0 . x∈P

It is well known (and easy to see) that N (P ) is a compact (n − 1)-dimensional submanifold of PX with singularities (warning: it is not a manifold with corners in general; it is smooth outside of a subset of codimension one). Also it is Legendrian with respect to the canonical contact structure on PX , though this fact will not be used explicitly in these lectures. Remark 1.2.1. If X = Rn and P ∈ P(Rn ) is convex, then this definition of the normal cycle coincides with the definition of the normal cycle from Section 1.1.12. Actually, the normal cycle can be defined for other classes of sets: sets of positive reach (which includes convex compact sets in the case X = Rn ), and subanalytic sets when X is a real analytic manifold (see Fu [32], which is based on [28, 29, 30, 31] and develops further [27, 62, 63]). An essentially equivalent notion of characteristic cycle was developed in [42] for subanalytic sets using a different approach. Below in this exposition we will assume for simplicity of exposition that X is oriented; this assumption can be easily removed. The orientation of X induces an orientation of the normal cycle of every subset. Definition 1.2.2. A map φ : P(X) → C is called a smooth valuation if there exist a measure μ on X and an (n − 1)-form ω on PX , both infinitely smooth, such that  φ(P ) = μ(P ) + ω N (P )

for any subset P ∈ P(X). Remark 1.2.3. This definition should be compared with Proposition 1.1.30. It can be shown that any translation-invariant convex valuation on Rn which is smooth in the sense of Definition 1.1.17 can be naturally extended to a broader class of sets: to compact sets of positive reach and also to relatively compact subanalytic

24

Chapter 1. New Structures on Valuations and Applications

subsets of Rn . This is done as follows: given a convex valuation φ ∈ Valsm (Rn ), let us represent it (non-uniquely) in the form  ω, φ(K) = a · vol(K) + N (K)

where ω is a smooth translation-invariant form. Then φ can be extended by the same formula to any subset from the above broader class; this extension is independent of the choice of the form ω and the constant a (see [9, Lemma 2.4.7]). It can be shown that every smooth valuation is a finitely additive functional in some precise sense [9]. Let us denote by V ∞ (X) the space of all smooth valuations. The space ∞ V (X) is the main object of study in what follows. Examples 1.2.4. (1) Any smooth measure on X is a smooth valuation. Indeed, take ω = 0 in Definition 1.2.2. (2) The Euler characteristic χ is also a smooth valuation. This fact is less obvious. In the current approach, it is a reformulation of a version of the Gauss–Bonnet formula due to Chern [26], who has constructed μ and ω to represent the Euler characteristic; his construction depends on the choice of a Riemannian metric on X. (3) The next example is very typical for integral geometry. Let X = CPn be the complex projective space. Let C Gr denote the Grassmannian of all complex projective subspaces of CPn of a fixed complex dimension k. It is well known that C Gr has a unique probability measure dE invariant under the group U (n + 1). Consider the functional  χ(P ∩ E) dE. φ(P ) = E∈C Gr

Then φ ∈ V ∞ (CPn ) —this follows, e.g., from Fu [31]. V ∞ (X) is naturally a Fr´echet space. Indeed, it is a quotient space of the direct sum of Fr´echet spaces M∞ (X) ⊕ Ωn−1 (PX ) by a closed subspace, where M∞ (X) denotes the space of infinitely smooth measures. The subspace of pairs (μ, ω) representing the zero valuation was described by Bernig and Br¨ ocker [23] in terms of a system of differential and integral equations. One can show [9] that smooth valuations form a sheaf. This means that (1) we have the natural restriction map V ∞ (U ) → V ∞ (V ) for any open subsets V ⊂ U ⊂ X; (2) given an open covering {Uα } of an open subset U , and φ ∈ V ∞ (U ) such that the restriction φ|Uα of φ to all Uα vanishes, one has φ = 0; (3) given an open covering {Uα } of an open subset U and φα ∈ V ∞ (Uα ) for any α such that φα |U α∩Uβ = φβ |U α∩Uβ for all α, β, there exists (uniquely by (2)) φ ∈ V ∞ (U ) such that φ|Uα = φα .

1.2. Valuations on manifolds

25

1.2.2 Canonical filtration on smooth valuations The space of smooth valuations carries a canonical filtration by closed subspaces. In this section we summarize its main properties without giving a precise definition, for which we refer to [9]. The important property of this filtration is that it partly allows to reduce the study of valuations on manifolds to the more familiar case of translation-invariant convex valuations. A more explicit geometric property of this filtration is given at the end of Section 1.2.5. Let us denote by Val(T X) the (infinite-dimensional) vector bundle over X such that its fiber over a point x ∈ X is equal to the space Val∞ (Tx X) of smooth theorem, it has a translation-invariant convex valuations on Tx X. By McMullen’s n (T X). grading by the degree of homogeneity: Val∞ (T X) = i=0 Val∞ i Theorem 1.2.5. There exists a canonical filtration of V ∞ (X) by closed subspaces V ∞ (X) = W0 ⊃ W1 ⊃ · · · ⊃ Wn ,

n = dim X,

n

such that the associated graded space i=0 Wi /Wi+1 is canonically isomorphic to the space of smooth sections C ∞ (X, Val∞ i (T X)). Remarks 1.2.6. (1) For i = n, the above isomorphism means that Wn coincides with the space of smooth measures on X. (2) For i = 0, the above isomorphism means that W0 /W1 is canonically isomorphic to the space of smooth functions C ∞ (X). The epimorphism V ∞ (X) → C ∞ (X) with kernel W1 is just the point evaluation map φ −→ [x −→ φ({x})]. Thus W1 consists precisely of valuations vanishing on all points. (3) Actually U → Wi (U ) defines a subsheaf Wi of the sheaf of valuations.

1.2.3 Integration functional Let Vc∞ (X) denote the subspace of V ∞ (X) of compactly supported valuations. (The definition is obvious: a valuation φ is said to have compact support if there exists a compact subset A ⊂ X such that the restriction φ|X\A is zero.) Clearly if X is compact then Vc∞ (X) = V ∞ (X). The space Vc∞ (X) carries a natural locally convex topology such that the natural imbedding to V ∞ (X) is continuous (however in general this is not a Fr´echet space, but rather a strict inductive limit of Fr´echet spaces; see [10, Section 5.1]). The integration functional  : Vc∞ (X) −→ C X

is defined by

 X

φ := φ(X).

26

Chapter 1. New Structures on Valuations and Applications

Formally speaking, φ(X) is not defined when X is not compact. The formal way to defineit is to choose first a large compact domain A containing the support of φ and set X φ := φ(A). Then one can show that this definition is independent of the large subset A. Moreover, X is a continuous linear functional.

1.2.4 Product operation on smooth valuations on manifolds and Poincar´e duality The product on smooth translation-invariant convex valuations, which was discussed in Section 1.1.7, can be extended to the case of smooth valuations on manifolds. We will describe below its main properties, and in Section 1.2.7 we will explain its intuitive meaning. However we present no construction of it in these notes. For the moment there are two different constructions of the product, both rather technical. The first one was done in several steps. Initially, the product was constructed by the author [8] on Rn (earlier the same construction was carried out in an even more specific situation [4] of convex valuations that are polynomial with respect to translations). Then this construction was extended by Fu and the author [18] to any smooth manifold: it was shown that the product can be defined locally choosing of a diffeomorphism of X with Rn and applying the above construction, and the main technical point was to show that the product is independent of the choice of this local diffeomorphism. The second and rather different construction of the product was given recently by Bernig and the author [16]. This construction describes the product of valuations directly in terms of the forms μ and ω defining the valuations; it uses the Rumin operator and some other standard operations on differential forms. Compared to the first construction, the second one has the advantage of being independent of extra structures on X (such as a coordinate system) and also some other technical advantages. However, it is less intuitive than the first one. We summarize basic properties of the product as follows. Theorem 1.2.7. There exists a canonical product V ∞ (X) × V ∞ (X) → V ∞ (X) such that (1) it is continuous; (2) it is commutative and associative; (3) the filtration W• is compatible with it: Wi · Wj ⊂ Wi+j , where we set Wk = 0 for k > n = dim X; (4) the Euler characteristic χ is the unit in the algebra V ∞ (X); (5) it commutes with restrictions to open and closed submanifolds. Thus V ∞ (X) is a commutative associative filtered unital algebra over C.

1.2. Valuations on manifolds

27

Let us also add that the point evaluation map V ∞ (X) → C ∞ (X) defined in Remark 1.2.6(2) is an epimorphism of algebras when C ∞ (X) is equipped with the usual pointwise product. An important property of the product is a version of Poincar´e duality. Consider the bilinear map V ∞ (X) × Vc∞ (X) −→ C  defined by (φ, ψ) → X φ · ψ. Theorem 1.2.8. This bilinear form is a perfect pairing. In other words, the induced map V ∞ (X) −→ (Vc∞ (X))∗ is injective and has a dense image with respect to the weak topology.

1.2.5 Generalized valuations and constructible functions Definition 1.2.9. The space of generalized valuations is defined as V −∞ (X) := (Vc∞ (X))∗ , equipped with the weak topology. Elements of this space are called generalized valuations. By Theorem 1.2.8 we have the canonical imbedding with dense image V ∞ (X) → V −∞ (X). Informally speaking, at least when X is compact, the space of valuations is essentially self-dual (up to completion). This imbedding also means that V −∞ (X) is a completion of V ∞ (X) in the weak topology. Every smooth valuation can be considered as a generalized one. The advantage of working with generalized valuations is that they contain the constructible functions (described below) as a dense subspace. This gives a completely different point of view on valuations which is often useful, especially on a heuristic level. Constructible functions have been studied quite extensively by methods of algebraic topology (sheaf theory; see the book [42, Ch. 9]). We will illustrate this below while discussing again the product on valuations, a Radontype transform, and the Euler–Verdier involution. Let us define the space of constructible functions on X. In the literature there are various, slightly different definitions of this notion, but the differences are technical rather than conceptual. For simplicity of exposition, we will assume in these notes, while talking about constructible functions, that X is a real analytic manifold. Definition 1.2.10. A function f : X → C on a real analytic manifold X is called constructible if it takes finitely many values and for any a ∈ C the level set f −1 (a) is subanalytic.

28

Chapter 1. New Structures on Valuations and Applications

For the definition of a subanalytic set, see [10, Section 1.2], or for more details [42, §8.2]. Constructible functions with compact support form a linear space which will be denoted by F(X). Moreover, F(X) is an algebra with pointwise product. An important property of constructible functions is that they also admit a normal cycle such that if P ∈ P(X) is subanalytic, then the normal cycle of the indicator function 1lP is equal to the normal cycle of P (see [32] and [42, Ch. 9]). Using this notion we define the map Ξ : F(X) −→ V −∞ (X) as follows. Let φ ∈ Vc∞ (X) be given by φ(P ) = μ(P ) + and ω. Then, for any f ∈ F(X),   f · dμ + ω. Ξ(f ), φ = X

 N (P )

ω, with smooth μ

N (f )

The map Ξ is well defined, i.e., it is independent of the particular choice of μ and ω representing φ. Moreover, Ξ is linear injective with dense image [10, Section 8.1]. To summarize, we have a large space of generalized valuations with two completely different dense subspaces, V ∞ (X) ⊂ V −∞ (X) ⊃ F(X).

(1.2.1)

Notice that, when X is compact, the image of the constant function 1 ∈ F(X) in V −∞ (X) coincides with the image of the Euler characteristic χ ∈ V ∞ (X). The two subspaces V ∞ (X) and F(X) are very different: thus, for connected X, their intersection is spanned by χ. While working with valuations it is useful to keep in mind the imbeddings (1.2.1). The role of constructible functions in the theory of valuations is somewhat analogous to the role of delta-functions in the classical theory of generalized functions (distributions). It is often instructive to compare various structures on valuations with their analogues on constructible functions. We will see several examples of this below. Here we will show how this works for the integration functional and the filtration W• .  It was shown in [10] that the integration functional X : Vc∞ → C extends uniquely by continuity in the weak topology to generalized valuations with compact support,  : Vc−∞ (X) −→ C. X

Let us restrict this functional to the subspace Fc (X) of constructible functions with compact support. It turns out that this restriction coincides with the integration with respect to the Euler characteristic; this operation is uniquely characterized by the property that, for any compact subanalytic subset P ⊂ X,  1lP = χ(P ). X

1.2. Valuations on manifolds

29

Let us consider now the filtration W• on V ∞ (X). Let Wi denote the closure of Wi in V −∞ (X) with respect to the weak topology. By [10] the restriction of Wi back to V ∞ (X) coincides with Wi , i.e., Wi ∩ V ∞ (X) = Wi . Consider the induced filtration on constructible functions, namely F(X) = F(X) ∩ W0 ⊃ F(X) ∩ W1 ⊃ · · · ⊃ F(X) ∩ Wn . It was shown in [10] that F(X) ∩ Wi consists of constructible functions whose support has codimension at most i. In particular, F(X) ∩ Wn consists of functions with discrete support.

1.2.6 Euler–Verdier involution Let us give another example of an application of the comparison with constructible functions. The space of constructible functions has a canonical linear involution, called the Verdier involution (see, e.g., [42]). In the special case of functions on Rn which are constructible in a narrower (polyhedral) sense, this involution has been known to convexity experts under the name of Euler involution. We will see that it extends naturally to valuations, and this extension will be called the Euler–Verdier involution. Here we will choose a sign normalization different from the standard one. Let us describe the Verdier involution σ (with a different sign convention) in the special case when a constructible function has the form 1lP , where P is a compact subanalytic submanifold with corners (the general case is not very far from this one using the linearity property of it). Then σ(1lP ) = (−1)n−dim P 1lint P , where int P denotes the relative interior of P . One has σ 2 = id. Theorem 1.2.11 ([10]). (1) The involution σ extends (uniquely) by continuity to V −∞ (X) in the weak topology. This extension is also denoted by σ. (2) σ preserves the class of smooth valuations and σ : V ∞ (X) → V ∞ (X) is a continuous linear operator (in the Fr´echet topology). (3) σ 2 = id. (4) σ : V ∞ (X) → V ∞ (X) is an algebra automorphism. (5) σ preserves the filtration W• , namely σ(Wi ) = Wi . (6) For any smooth homogeneous translation-invariant valuation φ on Rn one has (σφ)(K) = (−1)deg φ φ(−K), where deg φ denotes the degree of homogeneity of φ. (7) σ commutes with restrictions to open subsets (both for smooth and generalized valuations).

30

Chapter 1. New Structures on Valuations and Applications

Remark 1.2.12. Although the involution σ was defined above only on a real analytic manifold X, it can be defined on any smooth manifold as a continuous linear operator σ : V −∞ (X) → V −∞ (X). Then it satisfies the properties (2)–(7) of Theorem 1.2.11.

1.2.7 Partial product operation on generalized valuations In this section we discuss the promised intuitive meaning of the product on valuations. This interpretation was conjectured by the author [11] and proved rigorously by Bernig and the author [16]. It provides yet another example of the relevance of constructible functions to valuations. Recall again that we have the imbedding of smooth valuations into the generalized ones V ∞ (X) ⊂ V −∞ (X). One could try to extend the product on smooth valuations to V −∞ (X), say by continuity. Unfortunately, this is not possible. The situation here is much analogous to what is known in the classical theory of generalized functions (see, e.g., [41]). There the space of smooth functions C ∞ (X) is naturally imbedded into the larger space of generalized functions C −∞ (X), which is the completion of the former in the weak topology. The space C ∞ (X) has its usual pointwise product. However, this product does not extend to C −∞ (X) by continuity: for example, no rigorous way is known to take the square of the delta-function on X = R. Nevertheless it is still possible to define a partial product on C −∞ (X). This roughly means that one can define a product of two generalized functions whose “singularities” are disjoint. The precise technical condition is formulated in the language of the wave front sets of generalized functions in the sense of H¨ormander and Sato; we will not reproduce it here, but rather refer to [41]. This partial product is natural and enjoys some continuity properties [37, Ch. VI §3]. In the case of valuations we have the following result. Theorem 1.2.13 ([16]). There exists a partial product on V −∞ (X) extending the product on V ∞ (X). It is commutative and associative. We refer to [16] for the precise technical formulation when the partial product of two generalized valuations is defined. We notice only that the condition is also formulated in the language of wave front sets. Now we can try to restrict the partial product on generalized valuations to constructible functions and see what we get. The answer is very natural: we just get their pointwise product (under certain technical assumptions on the functions guaranteeing that their product in V −∞ (X) is well defined). More precisely, we have the following result. Theorem 1.2.14 ([16]). Let P , Q ⊂ X be compact submanifolds with corners which intersect transversally. Then the product of 1lP and 1lQ in the sense of generalized

1.2. Valuations on manifolds

31

valuations is well defined and is equal to 1lP ∩Q (notice that, under the transversality assumption, P ∩ Q is also a compact submanifold with corners). We did not give a formal definition of transversality of two submanifolds with corners. In the special case of submanifolds without corners, the definition is the usual one. In the general case, the precise definition is given in [16]. Notice only that any two compact submanifolds with corners can be made transversal to each other by applying to one of them a generic diffeomorphism which is arbitrarily close (in the C ∞ -topology) to the identity map.

1.2.8 A heuristic remark In this section we will make a general heuristic remark on valuations. In the next Section 1.2.9 we will show how it can be informally used in applications to integral geometry. Let ψ be a generalized valuation on a (say, real analytic) manifold X. Can we consider it as a finitely additive measure on X? The answer is “essentially” yes. This measure is partially defined: its value on a compact submanifold with corners or compact real analytic subset P , which is “in generic position” to ψ, is equal to X ψ · 1lP . Notice that, once the product is defined, the integral is defined too. When the valuation ψ is smooth, this integral has a very clear meaning, namely one has  ψ · 1lP = ψ(P ). (1.2.2) X

Let us prove the last identity. First, by the definition of 1lP , ψ(P ) = 1lP , ψ. But for any generalized valuation Φ with compact support one has  Φ · ψ. (1.2.3) Φ, ψ = X

To show this, let us observe that for fixed ψ both sides are continuous in Φ in the weak topology; hence, it suffices to show (1.2.3) for smooth Φ. But in this case this equality is just the definition of the imbedding V ∞ (X) → V −∞ (X). Let us specialize the above discussion to the case ψ = 1lA . Then we get a finitely additive partially defined measure P → χ(A ∩ P ), where P should be in a generic position with respect to A. Indeed,   1lA · 1lP = 1lA∩P = χ(A ∩ P ). P −→ X

X

1.2.9 A few examples of computation of the product in integral geometry In this section we give examples of computation of the product of valuations in the complex projective space CPn . These examples are very typical in integral

32

Chapter 1. New Structures on Valuations and Applications

geometry. We will use the heuristic discussion of the previous Section 1.2.8 since hopefully it will clarify the intuition behind the product in applications. Let now X = CPn with the Fubini–Study metric. Let us denote by C Gl the Grassmannian of l-dimensional complex projective subspaces of CPn . Clearly it is equal to the Grassmannian of (l + 1)-dimensional complex linear subspaces in Cn+1 . Let us consider the smooth U (n + 1)-invariant valuations  χ(K ∩ E) dE, (1.2.4) φl (K) := CG

l

C

where dE is the Haar measure on G normalized in some way (we do not care about normalization constants). We claim that  c · φl+m−n , l + m ≥ n, φl · φm = (1.2.5) 0, l + m < n, where c = 0 is a normalizing constant depending on normalizations of Haar measures and l, m, n. Let us give a heuristic proof of this equality. Using the discussion from the previous Section 1.2.8, we observe that   1lE dE (K), φl (K) = CG

l

where 1lE is considered as a generalized valuation. Hence   1lE · 1lF dE dF = φl · φm = (E,F )∈C Gl ×C Gm

CG

l×

CG

1lE∩F dE dF, m

where the last equality is due to Theorem 1.2.14. Since for generic projective subspaces E and F their intersection E ∩ F is a projective subspace of dimension l + m − n for l + m ≥ n and empty otherwise, it follows that   1lE∩F dE dF = c 1lM dM = c · φl+m−n . CG

l×

CG

CG

m

l+m−n

Thus the equality (1.2.5) is proved. Let us consider another important example of the product on CPn . We claim that the U (n + 1)-invariant valuation  Vi (K ∩ E) dE (1.2.6) K −→ CG

l

is equal to φl · Vi , where φl is defined by (1.2.4). First observe that, by (1.2.2), one has  Vi (K ∩ E) = 1lK∩E · Vi ,

1.2. Valuations on manifolds

33

 where 1lK∩E is considered as a generalized valuation and in the last expression is the integration functional (i.e., evaluation on the whole space CPn ). Now we use again Theorem 1.2.14 to write (under transversality assumptions) 1lK∩E = 1lK ·1lE . Thus   1lK∩E · Vi = 1lK · 1lE · Vi = (1lE · Vi )(K), where the last equality follows from the heuristic discussion of Section 1.2.8. Thus the valuation (1.2.6) is equal to    1lE · Vi dE = 1lE dE · Vi = φl · Vi , CG

CG

l

l

as claimed. Finally let us compute a generalization of the two previous examples. We claim that      Vi ( · ∩ E) dE · Vj ( · ∩ F ) dF = c · Vi+j ( · ∩ M ) dM, CG

l

CG

m

CG

l+m−n

(1.2.7) where c is a constant which can be computed explicitly. By the previous two examples of this section, Example 1.1.22 from Section 1.1.7, and using the associativity and the commutativity of the product, we see that the left-hand side of (1.2.7) is equal to (φl · Vi ) · (φm · Vj ) = (φl · φm ) · (Vi · Vj ) = c · φl+m−n · Vi+j = r.h.s. of (1.2.7). Thus (1.2.7) is proved.

1.2.10 Functorial properties of valuations We describe the operations of pull-back and push-forward on valuations under smooth maps of manifolds. These operations generalize the well-known operation of pull-back on smooth and constructible functions, the operation of push-forward on measures, and integration with respect to the Euler characteristic along the fibers (also called push-forward) on constructible functions. However, for the moment this is done rigorously only in several special cases of maps (say submersions and immersions). We believe, however, that these constructions can be extended to “generic” smooth maps as partially defined maps on valuations. The precise conditions under which the maps could be defined might be rather technical. For this reason we describe first the general picture heuristically. This picture should be considered as conjectural. Then we formulate several rigorous results with precise conditions under which one can define pull-back and push-forward on valuations. These special cases turn out to be sufficient to define rigorously the Radon-type transform on valuations (again under some conditions) in the next section. The results of this section have been obtained by the author in [14].

34

Chapter 1. New Structures on Valuations and Applications

Let us start with the heuristic picture. Denote by V (X) a space of valuations on a manifold X without specifying exactly the class of smoothness (smooth, generalized, or something else). Vc (X) denotes the subspace of V (X) of compactly supported valuations. Let f : X → Y be a smooth map of manifolds. There should exist a partially defined linear map, called push-forward, f∗ : Vc (X)  Vc (Y ), such that, for any nice subset P ⊂ Y , (f∗ φ)(P ) = φ(f −1 (P )).

(1.2.8)

Since (smooth) measures are contained in V (X), we can effect their push-forward in the sense of valuations. Clearly this operation should coincide with the classical push-forward of measures. It immediately follows from (1.2.8) that for the composition of maps we should have (f ◦ g)∗ = f∗ ◦ g∗ .

(1.2.9)

We expect that the following interesting property of push-forward f∗ holds. It should extend somehow to a partially defined map on generalized valuations. Hence f∗ can be restricted to a partially defined map on constructible functions; it should be defined on constructible functions which are “in generic position” with respect to the map f : X → Y . We expect that when f is a proper map (i.e., preimages of compact sets are compact), then on constructible functions f∗ coincides with integration with respect to the Euler characteristic along the fibers. Let us recall how this operation is defined assuming that X and Y are real analytic manifolds and f is a proper real analytic map. It is uniquely characterized by the following property: Let P ⊂ X be a subanalytic compact subset. Then (f∗ 1lP )(y) = χ(P ∩ f −1 (y)) for any point y ∈ Y . One can show that f∗ maps constructible functions to constructible ones. We refer to [42, Ch. 9] for further details. The push-forward should be related to the filtration on valuations in the following way: f∗ (Wi ) ⊂ Wi−dim X+dim Y . Also, f∗ should commute (up to a sign) with the Euler–Verdier involution. Let us now discuss the pull-back operation f ∗ : V (Y )  V (X), which should be a partially defined linear map in the opposite direction. Heuristically, f ∗ should be the dual map to f∗ (recall from Section 1.2.4 that Vc (X) and

1.2. Valuations on manifolds

35

V (X) are essentially dual to each other). The pull-back f ∗ should be a homomorphism of algebras of valuations (again, the product might be partially defined). We expect that f ∗ χ = χ. Also f ∗ should preserve the filtration f ∗ (Wi ) ⊂ Wi , and f ∗ should commute with the Euler–Verdier involution. Notice that, since in particular f ∗ (W1 ) ⊂ W1 , f ∗ induces a map between the quotients f ∗ : V (Y )/W1 −→ V (X)/W1 . But by Remark 1.2.6(2), V (Y )/W1 coincides with functions on Y of an appropriate class of smoothness. In particular, we should get a map f ∗ : C ∞ (Y ) −→ C −∞ (X). We expect that this is the usual pull-back on smooth functions, i.e., f ∗ (F ) = F ◦ f.

(1.2.10)

Now let us restrict f ∗ to constructible functions. We expect that it coincides again with the usual pull-back on constructible functions, which is defined by the same formula (1.2.10). Finally, consider the restriction of f ∗ to (say smooth) measures on Y . In classical measure theory, the operation of pull-back of a measure does not exist. Nevertheless, it is possible to define such a pull-back as a valuation, at least under appropriate technical conditions on the map f . Let μ be a smooth measure on Y . Then, leaving all the technicalities aside, one should have  χ(P ∩ f −1 (y)) dμ(y). (f ∗ μ)(P ) = y∈Y

In particular, if f : X → Y is a linear projection of vector spaces and P ⊂ X is a convex compact subset, then (f ∗ μ)(P ) = μ(f (P )) is the measure of the projection of P . Now let us describe several rigorous results which will be used later. Let f : X → Y be a smooth map. Case 1: Assume that f is a closed imbedding. Then the obvious restriction map V ∞ (Y ) → V ∞ (X) defines the pull-back map f ∗ , which is a linear continuous operator. Dualizing it, we get the push-forward map f∗ : V −∞ (X) −→ V −∞ (Y ), which is a linear continuous operator (in the weak topology). Notice that in this situation f∗ does not preserve the class of smooth valuations. It was shown in [14] that in this case f∗ (1lP ) = 1lf (P ) for any compact submanifold with corners P ⊂ X. It was also shown that f ∗ can be extended to a

36

Chapter 1. New Structures on Valuations and Applications

partially defined map V −∞ (Y )  V −∞ (X) such that if Q ⊂ Y is a compact submanifold with corners which is transversal to X, then f ∗ 1lQ is well defined in the sense of valuations and is equal to 1lX∩Q , i.e., the pull-back on valuations is compatible with the pull-back on constructible functions. Case 2: Assume that f is a proper submersion. Let us define the push-forward f∗ : Vc∞ (X) → Vc∞ (Y ) by (f∗ φ)(P ) = φ(f −1 (P )) for any compact submanifold with corners P ⊂ Y . Notice that in this case f −1 (P ) is a compact submanifold with corners, and f∗ φ is indeed a smooth valuation. The map constructed is linear and continuous. Taking the dual map, we define the pull-back map f ∗ : V −∞ (Y ) −→ V −∞ (X). It was shown in [14] that in this case for any compact submanifold with corners P ⊂ Y one has f ∗ (1lP ) = 1lP ◦ f = 1lf −1 (P ) . It was also shown that the pushforward f∗ extends to a partially defined map on generalized valuations. However, its compatibility with integration with respect to the Euler characteristic along the fibers was proved only under rather unpleasant restrictions on the class of constructible functions.

1.2.11 Radon transform on valuations on manifolds In this section we combine the product, pull-back, and push-forward on valuations to define a Radon-type transform on them. Before we introduce this notion, it is instructive to recall the general Radon transform on smooth functions following Gelfand, and less classical but still known Radon transform on constructible functions. These two completely different transforms can be considered as special cases of the general Radon transform on valuations. In our opinion, this is the most interesting property of the new Radon transform on valuations. Definition 1.2.15. A double fibration is a triple of smooth manifolds X, Y , and Z with two submersive maps p q X ←− Z −→ Y such that the map p × q : Z −→ X × Y is a closed imbedding. To define a general Radon transform on smooth functions, let us fix a double fibration as above and an infinitely smooth measure γ on Z. Let us also assume that q : Z → Y is proper. The Radon transform is the operator Rγ : Cc∞ (X) → M∞ (Y ) (where M∞ (Y ) denotes the space of smooth measures) defined by Rγ f := q∗ (γ · p∗ f ),

(1.2.11)

where p∗ f = f ◦ p is the usual pull-back on smooth functions, the product is just the usual product of a measure by a function, and q∗ is the usual push-forward on measures. Notice that all classical Radon transforms on smooth functions have

1.2. Valuations on manifolds

37

such a form. For example, let us take X = Rn , Y the Grassmannian of affine k-dimensional subspaces, and Z the incidence variety, i.e., Z = {(x, E) ∈ X × Y | x ∈ E}. Let γ be a Haar measure on Z invariant under the group of all isometries of Rn . Then Rγ is the classical Radon transform, given by integration of a function on Rn over all affine k-dimensional subspaces. There is a very extensive literature on this subject; see, e.g., [34, 35, 40]. Let us recall the Radon transform with respect to the Euler characteristic on constructible functions. It was studied for the real projective space and a somewhat more restrictive class of constructible functions by Khovanskii and Pukhlikov [46]; their work has been motivated by the earlier work of Viro [61] on the Radon transform on complex constructible functions on complex projective spaces. We will discuss and generalize the Khovanskii–Pukhlikov result in the next section. For subanalytic constructible functions and other spaces, the Radon transform with respect to the Euler characteristic was studied by Schapira [57]. Thus let p

q

X ←− Z −→ Y be a double fibration of real analytic spaces with real analytic maps p and q. We assume again that q is proper. Let us denote by F(X) the space of constructible functions, as defined in Section 1.2.5. Then one defines the Radon transform R : F(X) → F(Y ) by Rf := q∗ p∗ (f ),

(1.2.12)

where p∗ denotes the usual pull-back on (constructible) functions, and q∗ is integration with respect to the Euler characteristic along the fibers of q. With these preliminaries, let us introduce the Radon transform on valuations. We fix a double fibration as above with the map q being proper. Let us fix a smooth valuation γ ∈ V ∞ (Z). We define the Radon transform on valuations Rγ : V ∞ (X) → V −∞ (Y ) by Rγ (φ) = q∗ (γ · p∗ φ), where p∗ and q∗ are the pull-back and push-forward on valuations, respectively, and the product with γ is taken in the sense of valuations. It was shown in [14] that Rγ is a well-defined continuous linear operator. Let us comment on some of the technical difficulties in this construction. Usually p∗ φ is not a smooth valuation, though φ is. Thus we have to multiply the smooth valuation γ by the non-smooth p∗ φ. This is always possible in the class of generalized valuations, but the product is not a smooth valuation. Next we have to take the push-forward of this generalized valuation. The push-forward of a generalized valuation under a general proper submersion is not always defined; it

38

Chapter 1. New Structures on Valuations and Applications

is so only under a rather technical condition of “generic position” of “singularities” of the valuation with respect to the map q. Fortunately, this technical condition is satisfied for valuations of the form γ · p∗ φ with smooth φ. It was also shown in [14] that under extra assumptions on the double fibration, the image Rγ (V ∞ (X)) is contained in smooth valuations. Also under a similar extra assumption Rγ can be extended uniquely by continuity in the weak topology to generalized valuations V −∞ (X). An example satisfying both assumptions will be considered in the next section. Let us discuss now the relation of the new Radon transform on valuations to the classical Radon transforms discussed above in this section. First let us assume that the valuation γ ∈ V ∞ (Z) is in fact a smooth measure considered as a smooth valuation. Then the Radon transform Rγ : V ∞ (X) −→ V −∞ (Y ) vanishes on W1 ⊂ V ∞ (X). Indeed p∗ (W1 ) ⊂ W1 and γ · W1 = 0, since γ is a measure. Hence Rγ factorizes (uniquely) via the quotient V ∞ (X)/W1 = C ∞ (X). Notice also that in this case Rγ takes values in measures, in fact in infinitely smooth ones. Hence we get a map C ∞ (X) → M∞ (Y ). It was shown in [14] that this map coincides with the classical Radon transform Rγ defined by (1.2.11). Let us consider another extremal case of Rγ with γ = χ being the Euler characteristic. In this case our discussion will be less rigorous. First assume that Rγ extends naturally to a partially defined map on generalized valuations V −∞ (X)  V −∞ (Y ). We expect that its restriction to the class of constructible functions coincides with the Radon transform with respect to the Euler characteristic defined previously by (1.2.12). This result was proved rigorously in [14] in very special circumstances. It is desirable to make the result rigorous under more general assumptions.

1.2.12 Khovanskii–Pukhlikov-type inversion formula for the Radon transform on valuations on RPn Let us consider the Radon-type transform on valuations in the following special case. Let X = RPn be the real projective space, i.e., the manifold of lines in Rn+1 passing through the origin. Let Y = RPn∨ be the dual projective space, i.e., the manifold of linear hyperplanes in Rn+1 . Let Z ⊂ X × Y be the incidence variety   Z := (l, E) ∈ RPn × RPn∨ | l ⊂ E . We have the double fibration p

q

RPn ←− Z −→ RPn∨ , where p and q are the obvious projections. All the manifolds and maps are real analytic.

1.2. Valuations on manifolds

39

We consider the Radon transform Rχ : V ∞ (RPn ) −→ V −∞ (RPn∨ ) with the kernel γ = χ being the Euler characteristic on Z. In this case R χ = q ∗ p∗ . It was shown in [14] that the image of this transformation is contained in smooth valuations, and that Rχ : V ∞ (RPn ) → V ∞ (RPn∨ ) is continuous. Moreover, this operator extends (uniquely) to a continuous linear operator, also denoted by Rχ , on generalized valuations equipped, as usual, with the weak topology: Rχ : V −∞ (RPn ) −→ V −∞ (RPn∨ ). It was shown in [14] that Rχ is invertible for odd n, and for even n its kernel consists precisely of multiples of the Euler characteristic. In both cases there is an explicit inversion formula (in the latter case, up to a multiple of the Euler characteristic); it generalizes and was motivated by the Khovanskii–Pukhlikov inversion formula for constructible functions [46]. In order to state the result, let us consider the analogous operator in the opposite direction, Rtχ : V −∞ (RPn∨ ) −→ V −∞ (RPn ), namely

Rtχ := p∗ q ∗ .

Theorem 1.2.16 ([14]). For any generalized valuation φ ∈ V −∞ (RPn ) one has    1 n−1 t n−1 Rχ Rχ (φ) = φ + (−1) −1 φ · χ. (1.2.13) (−1) 2 RPn Let us say a few words about the proof of this theorem. After all the operators involved were defined, the next technically non-trivial step was to show that the restriction of Rχ to a rather special class of constructible functions, which is still dense in V −∞ (RPn ), coincides with the Radon transform with respect to the Euler characteristic on constructible functions; also, an analogous result holds for Rtχ . Then Theorem 1.2.16 follows immediately by continuity from the Khovanskii– Pukhlikov inversion formula for constructible functions, which claims precisely the identity (1.2.13) for such functions in place of φ.

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[13] Alesker, S.: Plurisubharmonic functions on the octonionic plane and Spin(9)invariant valuations on convex sets. J. Geom. Anal. 18 (2008), no. 3, 651–686. [14] Alesker, S.: Valuations on manifolds and integral geometry. Geom. Funct. Anal. 20 (2010), no. 5, 1073–1143. [15] Alesker, S.: A Fourier type transform on translation invariant valuations on convex sets. Israel J. Math. 181 (2011), 189–294. [16] Alesker, S. and Bernig, A.: The product on smooth and generalized valuations. Amer. J. Math. 134 (2012), no. 2, 507–560. [17] Alesker, S. and Bernstein, J.: Range characterization of the cosine transform on higher Grassmannians. Adv. Math. 184 (2004), no. 2, 367–379. [18] Alesker, S. and Fu, J. H. G.: Theory of valuations on manifolds, III. Multiplicative structure in the general case. Trans. Amer. Math. Soc. 360 (2008), no. 4, 1951–1981. [19] Bernig, A.: A Hadwiger-type theorem for the special unitary group. Geom. Func. Anal. 19 (2009), 356–372. [20] Bernig, A.: Integral geometry under G2 and Spin(7). Israel J. Math. 184 (2011), 301–316. [21] Bernig, A.: Invariant valuations on quaternionic vector spaces. J. Inst. Math. Jussieu 11 (2012), 467–499. [22] Bernig, A.: Algebraic integral geometry. In: Global Differential Geometry (C. B¨ ar, J. Lohkamp and M. Schwarz, eds.), 107–145, Springer Proceedings in Math. Vol. 17, Springer, Berlin, 2012. [23] Bernig, A. and Br¨ocker, L.: Valuations on manifolds and Rumin cohomology. J. Differential Geom. 75 (2007), 433–457. [24] Bernig, A. and Fu, J. H. G.: Convolution of convex valuations. Geom. Dedicata 123 (2006), 153–169. [25] Bernig, A. and Fu, J. H. G.: Hermitian integral geometry. Ann. of Math. 173 (2011), 907–945. [26] Chern, S.-S.: On the curvatura integra in a Riemannian manifold. Ann. of Math. 46 (1945), 674–684. [27] Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418– 491. [28] Fu, J. H. G.: Curvature measures and generalized Morse theory, J. Differential Geom. 30 (1989), 619–642. [29] Fu, J. H. G.: Monge–Amp`ere functions, I. Indiana Univ. Math. J. 38 (1989), no. 3, 745–771.

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[30] Fu, J. H. G.: Monge–Amp`ere functions, II. Indiana Univ. Math. J. 38 (1989), no. 3, 773–789. [31] Fu, J. H. G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39 (1990), no. 4, 1115–1154. [32] Fu, J. H. G.: Curvature measures of subanalytic sets. Amer. J. Math. 116 (1994), no. 4, 819–880. [33] Fu, J. H. G.: Structure of the unitary valuation algebra. J. Differential Geom. 72 (2006), no. 3, 509–533. [34] Gelfand, I. M., Graev, M. I. and Ro¸su, R.: The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. J. Operator Theory 12 (1984), no. 2, 359–383. ˇ [35] Gelfand, I. M., Graev, M. I. and Sapiro, Z. Ja.: Integral geometry on k-dimensional planes. (Russian) Funkcional. Anal. i Priloˇzen 1 (1967), 15–31. [36] Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. [37] Guillemin, V. and Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, no. 14. American Mathematical Society, Providence, R.I., 1977. [38] Hadwiger, H.: Translationsinvariante, additive und stetige Eibereichfunktionale. (German) Publ. Math. Debrecen 2 (1951), 81–94. [39] Hadwiger, H.: Vorlesungen u ¨ber Inhalt, Oberfl¨ ache und Isoperimetrie. (German) Springer-Verlag, Berlin-G¨ ottingen-Heidelberg, 1957. [40] Helgason, S.: The Radon Transform. Second edition. Progress in Math. 5. Birkh¨auser Boston, Inc., Boston, MA, 1999. [41] H¨ormander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Reprint of the second (1990) edition. Classics Math. Springer-Verlag, Berlin, 2003. [42] Kashiwara, M. and Schapira, P.: Sheaves on Manifolds. Grundlehren Math. Wiss. 292. Springer-Verlag, Berlin, 1990. [43] Klain, D. A.: A short proof of Hadwiger’s characterization theorem. Mathematika 42 (1995), no. 2, 329–339. [44] Klain, D. A.: Even valuations on convex bodies. Trans. Amer. Math. Soc. 352 (2000), no. 1, 71–93. [45] Klain, D. A. and Rota, G.-C.: Introduction to Geometric Probability. Lezioni Lincee. Cambridge University Press, Cambridge, 1997.

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[46] Khovanskii, A. G. and Pukhlikov, A. V.: Finitely additive measures of virtual polyhedra. (Russian) Algebra i Analiz 4 (1992), no. 2, 161–185; translation in St. Petersburg Math. J. 4 (1993), no. 2, 337–356. [47] Ludwig, M.: Projection bodies and valuations. Adv. Math. 172 (2002), no. 2, 158–168. [48] Ludwig, M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119 (2003), no. 1, 159–188. [49] Ludwig, M.: Intersection bodies and valuations. Amer. J. Math. 128 (2006), no. 6, 1409–1428. [50] Ludwig, M. and Reitzner, M.: A characterization of affine surface area. Adv. Math. 147 (1999), no. 1, 138–172. [51] Ludwig, M. and Reitzner, M.: A classification of SL(n) invariant valuations. Ann. of Math. 172 (2010), no. 2, 1219–1267. [52] McMullen, P.: Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. 35 (1977), no. 1, 113–135. [53] McMullen, P.: Continuous translation-invariant valuations on the space of compact convex sets. Arch. Math. (Basel) 34 (1980), no. 4, 377–384. [54] McMullen, P.: Valuations and dissections. In: Handbook of Convex Geometry, Vol. B, 933–988, North-Holland, Amsterdam, 1993. [55] McMullen, P. and Schneider, R.: Valuations on convex bodies. Convexity and its applications, 170–247, Birkh¨ auser, Basel, 1983. [56] Santal´o, L. A.: Integral Geometry and Geometric Probability. Encyclopedia Math. Appl., 1. Addison-Wesley Publishing Co., Reading, Mass.-LondonAmsterdam, 1976. [57] Schapira, P.: Tomography of constructible functions. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Paris, 1995), 427–435, Lecture Notes in Comput. Sci. 948, Springer, Berlin, 1995. [58] Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia Math. Appl., 44. Cambridge University Press, Cambridge, 1993. [59] Schneider, R.: Simple valuations on convex bodies. Mathematika 43 (1996), no. 1, 32–39. [60] Schneider, R. and Schuster, F. E.: Rotation equivariant Minkowski valuations. Int. Math. Res. Not. 2006, Art. ID 72894, 20 pp. [61] Viro, O.: Some integral calculus based on Euler characteristic. In: Topology and Geometry – Rohlin Seminar, 127–138, Lecture Notes in Math. 1346, Springer, Berlin, 1988.

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[62] Wintgen, P.: Normal cycle and integral curvature for polyhedra in Riemannian Manifolds. In: Differential Geometry (G. Soos and J. Szenthe, eds.), NorthHolland, Amsterdam, 1982. [63] Z¨ahle, M.: Approximation and characterization of generalised Lipschitz– Killing curvatures. Ann. Global Anal. Geom. 8 (1990), no. 3, 249–260.

Chapter 2

Algebraic Integral Geometry Joseph H. G. Fu Introduction Recent work of S. Alesker has catalyzed a flurry of progress in Blaschkean integral geometry and opened the prospect of further advances. By this term we understand the circle of ideas surrounding the kinematic formulas (Theorem 2.1.6 below), which express related fundamental integrals relating to the intersections of two subspaces K, L ⊂ Rn in general position in terms of certain “total curvatures” of K and L separately. The purest form of this study concerns the integral geometry of isotropic spaces (M, G), i.e., Riemannian manifolds M equipped with a subgroup G of the full isometry group that acts transitively on the tangent sphere bundle SM . That kinematic formulas exist in this generality was first established in [36]. However, it was not until the discovery by Alesker of the rich algebraic structure on the space of convex valuations, and his extension of these notions to arbitrary smooth manifolds, that the determination of the actual formulas has become feasible. The key fact is the Fundamental Theorem of Algebraic Integral Geometry, or FTAIG (Theorems 2.3.16 and 2.4.15 below), which relates the kinematic formulas for an isotropic space (M, G) to the finite-dimensional algebra of G-invariant valuations on M . The classical approach to integral geometry relies on the explicit calculation of multiple integrals (the template method described in Section 2.1.4 below) and intricate numerical identities. Even apart from the unsatisfyingly ad hoc nature of these arguments, they are also notoriously difficult in practice: for example, even such an eminently skilled calculator as S.-S. Chern [33] published fallacious values for some basic integral geometric constants (cf. [52]). While Alesker theory has not entirely expunged such considerations from integral geometry, it has provided an array of “soft” tools that give a structural rationale and thereby a practical method of checking. As a result, many old mysteries have been resolved and more new ones

© Springer Basel 2014 S. Alesker, J.H.G. Fu, Integral Geometry and Valuations, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0874-3_2

47

48

Chapter 2. Algebraic Integral Geometry

uncovered. In a general way this development was foreseen by G.-C. Rota [49], who famously characterized integral geometry as “continuous combinatorics”.

About these notes These notes represent a revision of the notes for my Advanced Course on Integral Geometry and Valuation Theory at the Centre de Recerca Matem` atica (Barcelona) in September 2010. My general aim was to convey an intuitive working knowledge of the state of the art in integral geometry in the wake of the Alesker revolution. As such, I have tried to include exact formulas wherever possible, without getting bogged down in technical details in the proofs and formal statements. As a result some of the proofs and statements are incomplete or informal, e.g., sometimes concluding the discussion once I feel the main points have been made, or when I do not see how to bring it to an end in a quick and decisive fashion. Some of the unproved assertions here have been treated more fully in the recent paper [26]. I intend to continue to develop this account to the point where it is reliable and complete, so comments on any aspect (especially corrections) either in person or by email will be welcome. Of course the topics included reflect my personal understanding and limitations. For example, there is nothing here about Bernig’s determination of the integral geometry of       n C , SU (n) , R8 , Spin(7) , and R7 , G2 . Bernig’s excellent account [21] discusses these topics and offers a different viewpoint on the theory as a whole. There are (at least) two other meanings covered by the term “integral geometry” that we will not discuss here. First is the integral geometry initiated by Gelfand, having to do with the Radon transform and its cousins. Remarkably, recent work of Alesker suggests that the concept of valuation may prove to be the correct formal setting also for this ostensibly separate study. Second is a raft of analytic questions which may be summarized as: which spaces and subspaces are subject to the kinematic formulas? They are known to include disparate classes coming from algebraic geometry, hard analysis, and convexity, and all seem to share certain definite aspects of a Riemannian space [38, 39]. The Exercises include assertions that I have not completely thought through, so they may be much harder or much easier than I think, and also may be false. Some statements I have labeled as Problems, which means that I believe they represent serious research projects.

Acknowledgments I am extremely grateful to the Centre de Recerca Matem` atica, and to the organizers E. Gallego, X. Gual, G. Solanes, and E. Teufel, for their invitation to lecture on

2.1. Classical integral geometry

49

this subject in the Advanced Course on Integral Geometry and Valuation Theory, which provided the occasion for these notes. In addition, I would like to thank A. Bernig and S. Alesker for extremely fruitful collaborations and discussions over the past few years as this material was worked out.

Notation and conventions Classic texts on the subject of integral geometry include [28, 43, 49, 56, 57]. The volumes of the k-dimensional unit ball and the k-dimensional unit sphere are denoted respectively by k

π2 , Γ(1 + k2 )   αk := k + 1 ωk+1 . ωk :=

Grk (V ) denotes the Grassmannian of all k-dimensional vector subspaces of the real vector space V and Grk (V ) denotes the corresponding space of affine subspaces. Given a group G acting linearly on a vector space V , we denote by G := GV the associated group of affine transformations. If G acts transitively by isometries on a Riemannian manifold M , we will usually normalize the Haar measure dg on G so that   dg g : go ∈ S = vol(S) (2.0.1) for S ⊂ M , where o ∈ M is an arbitrarily chosen base point. If S is a subset of a vector space, then S denotes its linear (or sometimes affine) span. Sometimes |S| will denote the volume of S, with the dimension understood from the context. In some cases πE will denote the orthogonal projection onto E, and in others it will be the projection onto the factor E of a cartesian product. For simplicity, I will generally endow all vector spaces with Euclidean structures and all manifolds with Riemannian metrics. This device carries the advantage that the dual spaces and cotangent vectors can then be identified with the original spaces and tangent vectors. Unfortunately, it also obscures the natural generality of some of the constructions, and can lead to outright misconceptions (e.g., in applications to Finsler geometry) if applied with insufficient care.

2.1 Classical integral geometry We consider Rn with its usual Euclidean structure, together with the group SO(n) := SO(n)  Rn

50

Chapter 2. Algebraic Integral Geometry

of orientation-preserving isometries. Let K = K(Rn ) denote the metric space of all compact convex subsets A ⊂ Rn , endowed with the Hausdorff metric     (2.1.1) d A, B := inf r ≥ 0 : A ⊂ Br , B ⊂ Ar , where Br is the euclidean ball of radius r and   Ar := x ∈ Rn : dist(x, A) ≤ r . We denote by Ksm = Ksm (Rn ) the dense subspace of convex subsets A with nonempty interior and smooth boundary, and such that all principal curvatures k1 , . . . , kn−1 are positive at every point of ∂A.

2.1.1

Intrinsic volumes and Federer curvature measures

Our starting point is Theorem 2.1.1 (Steiner’s formula). If A ∈ K, then n    ωn−i μi (A) rn−i , vol Ar =

r ≥ 0.

(2.1.2)

i=0

The functionals μi (thus defined) are called the intrinsic volumes, and are equal up to scale to the Quermassintegrale introduced by Minkowski (cf. [28, p. 49]). A simple geometric argument shows that A ∈ K, dim A = j =⇒ μj (A) = volj (A). It is easy to prove (2.1.2) if A ∈ Ksm . In this case   Ar = A ∪ expA ∂A × [0, r] , where

  expA x, t := x + tnA (x)

(2.1.3)

(2.1.4) (2.1.5)

is the Gauss map. It is clear that expA gives a diffeomorphism and nA : ∂A → S ∂A × (0, ∞) → Rn  A: in fact the inverse map may be written in terms of     x = πA expA (x, t) , t = δA expA (x, t) , n−1

where πA : Rn → A is the nearest point projection and δA is the distance from A. Clearly,   (2.1.6) Dx,t0 expA v + c∂t = v + t0 Lx (v) + cnA (x) for v ∈ Tx ∂A, where Lx : Tx ∂A → Tx ∂A = TnA (x) S n−1 ⊂ Rn is the Weingarten map. Thus the area formula gives   r   d voln−1 x dt det IdTx ∂A +tLx vol (Ar ) = vol (A) + ∂A

0

2.1. Classical integral geometry

51 

 d voln−1 x

= vol (A) + ∂A

= vol (A) +

r

n−1  j=0

dt 0

j+1

r j+1



n−1 



1 + tkj



j=1

  σj k1 , . . . , kn−1 d voln−1 x, ∂A

where the kj are the principal curvatures and σj is the jth elementary symmetric polynomial. In particular,  1 σn−i−1 (k1 , . . . , kn−1 ) d voln−1 x (2.1.7) μi (A) = (n − i)ωn−i ∂A in this case. Note that the μi are independent of the ambient dimension, i.e., if j : Rn → RN is a linear isometry, then μi (j(A)) = μi (A). Observe that μn (A) = |A|; μ0 (A) = 1, i

μi (tA) = t μi (A), μi (x + A) = μi (A),

(by Gauss-Bonnet); for 0 = t ∈ R; for x ∈ Rn .

A modification of this approach establishes (2.1.2) for general A ∈ K. Note that the function δA := dist( · , A) is C 1 when restricted to the complement of A, with x − πA (x) . (2.1.8) ∇δA (x) = |x − πA (x)| 1,1 Since πA is clearly Lipschitz (with constant 1), in fact δA ∈ Cloc (Rn  A), and the n n n n−1 given by map ΠA : R  A → SR := R × S   (2.1.9) ΠA (x) := πA (x), ∇δA (x)

is locally Lipschitz. In fact, for fixed r > 0 the restriction of ΠA to ∂Ar is a biLipschitz homeomorphism to its image. Furthermore, this image is independent of r, and these maps commute with the obvious projections between the various ∂Ar . Precisely, the image is the normal cycle of A,      (2.1.10) N (A) := x, v ∈ SRn : v, x − y ≥ 0 for all y ∈ A , which thus may be regarded as an oriented Lipschitz submanifold of SRn of dimension n − 1. Now the exponential map exp : SRn × R → Rn , exp(x, v; t) := x + tv, yields a locally bi-Lipschitz homeomorphism N (A) × (0, ∞) → Rn  A, with Ar  A = exp(N (A) × (0, r]). Thus the volume of Ar may be expressed as  exp∗ d vol, (2.1.11) vol (Ar ) = vol(A) + N (A)×(0,r]

52

Chapter 2. Algebraic Integral Geometry

where

      exp∗ d vol = d x1 + tv1 ∧ · · · ∧ d xn + tvn

(2.1.12)

is a differential form of degree n on SRn . Since d vol is invariant and exp is covariant under the action of SO(n), this form is again SO(n)-invariant. In order to understand this form it is therefore enough to evaluate it at the special point (0, en ) ∈ SRn , where and α := Thus

n

vn = 1,

i=1

v1 = · · · = vn−1 = 0,

dvn = 0,

dxn = α,

vi dxi is the invariant canonical 1-form of the sphere bundle SRn .

    exp∗ (d vol)(0,en ) = d x1 + tv1 ∧ · · · ∧ d(xn−1 + tvn−1 ) ∧ α + dt  n−1    i t κn−i−1 ∧ α + dt = i=0

for some invariant forms κi ∈ Ωn−1,SO(n) (SRn ) of degree n − 1, and (2.1.11) may be expressed as n−1  ri+1  κn−i−1 . (2.1.13) vol(Ar ) = vol(A) + i + 1 N (A) i=0 Thus, for general A ∈ K, 1 μi (A) = (n − i)ωn−i

 κi .

(2.1.14)

N (A)

The formula (2.1.14) is of course a direct generalization of (2.1.7), and the latter may be computed directly from the former by pulling back via the diffeomorphism ¯ A (x) := (x, nA (x)). n ¯ A : ∂A → N (A), n It is natural to express the intrinsic volumes as the “complete integrals” of the Federer curvature measures  1 (E) := κi , i = 0, . . . , n − 1, (2.1.15) ΦA i (n − i)ωn−i N (A)∩πR−1 n (E)   (2.1.16) ΦA n (E) := vol A ∩ E . These measures satisfy n    −1 n−i vol Ar ∩ πA (E) = ωn−i ΦA . i (E)r

(2.1.17)

i=0

The discussion above goes through also in case the compact set A is only semiconvex, i.e., there is r0 > 0 such that if δA (x) < r0 , then there exists a unique πA (x) := p ∈ A such that δA (x) = |x − p| (it is necessary of course to restrict to r < r0 throughout). The supremum of such r0 is called the reach of A [34]. Note that reach(A) = ∞ if and only if A is convex. In this more general setting, the Federer curvature measures ΦA i will generally be signed.

2.1. Classical integral geometry

53

2.1.2 Other incarnations of the normal cycle Consider also    (A) := (x, v) ∈ T Rn : x ∈ A, v, x − y ≥ 0 for all y ∈ A , N    (A) ∩ Rn × B1 , N1 (A) := N       ∗ (A) := x, ξ ∈ T ∗ Rn : x ∈ A, ξ, x − y ≥ 0 for all y ∈ A , N    (A) , N ∗ (A) := (x, [ξ]) ∈ S ∗ Rn : (x, ξ) ∈ N

(2.1.18) (2.1.19) (2.1.20) (2.1.21)

where S ∗ Rn is the cosphere bundle. The last two are in some sense the correct objects to think about, since they behave naturally under linear changes of variable.

2.1.3 Crofton formulas The intrinsic volumes occur naturally in certain questions of geometric probability. The starting point is Crofton’s formula. In the convex case this may be stated as  πE (A) dE. (2.1.22) μn−1 (A) = Grn−1

Here the codimension 1 intrinsic volume is half of the perimeter and dE is a Haar measure that will be chosen below. This may be proved via the area formula. Fixing E ∈ Grn−1 with unit normal vector v = vE , and x ∈ ∂A, the Jacobian determinant of the restriction of DπE to Tx ∂A is det DπE |Tx ∂A = |v · nA (x)|. Hence



πE (A) dE = 1 2 Grn−1  =





v · nA (x) dx

dE Grn−1

∂A



dx S n−1

(2.1.23)

∂A

|v · u| dv

= μn−1 (A),

(2.1.24) (2.1.25)

where u ∈ S n−1 is an arbitrarily chosen unit vector, provided the Haar measures dE and dv are normalized appropriately. Observe that  πE (Ar ) = πE (A) r . Hence, by Steiner’s formula, d μn−1 (Ar ) dr   d πE (A) dE =c r dr Grn−1

μn−2 (Ar ) = c

54

Chapter 2. Algebraic Integral Geometry  =c 

Grn−1



Grn−1

=c =c  =

d  πE (A) r dE dr μn−2 (πE (Ar )) dE    dE voln−2 πF (Ar ) dF

Grn−1

Grn−2 (Rn )

Grn−2 (E)

voln−2 (πF (Ar )) dF

with conveniently normalized Haar measure dF on Grn−2 , since πF ◦ πE = πF when F is a subspace of E. Setting r = 0 and continuing in this way, Proposition 2.1.2. If the Haar measure dG is appropriately normalized, then  volk (πG ( · )) dG, k = 0, . . . , n − 1. (2.1.26) μk = Grk

This can also be written in the following equivalent form. Let Grn−k denote the space of affine planes of dimension n − k in Rn . Then    ¯ dH, ¯ μk (A) = χ A∩H k = 0, . . . , n − 1, (2.1.27) Grn−k

¯ is the Haar measure obtained by where χ is the Euler characteristic, and dH viewing Grn−k as the total space of the tautological bundle over Grk and taking the product of dG from Proposition 2.1.2 with the Lebesgue measure on the fibers. Exercise 2.1.3. Deduce that    m μk πE (A) dE μk (A) = ck  Grm   ¯ dH. ¯ μk−j A ∩ H = djk

(m ≥ k) (n ≥ k ≥ j)

Grn−j j Is it possible to choose the normalizations of the Haar measures so that cm k , dk ≡ 1?

2.1.4 The classical kinematic formulas It follows immediately from equation (2.1.2) that the μk are continuous, translation-invariant and SO(n)-invariant valuations, i.e., (1) A, B, A ∪ B ∈ K =⇒ μk (A ∪ B) + μk (A ∩ B) = μk (A) + μk (B). (2) μk (x + A) = μk (A) for all x ∈ Rn . (3) Ai → A in the Hausdorff metric =⇒ μk (Ai ) → μk (A).

2.1. Classical integral geometry

55

(4) μk (gA) = μk (A) for all g ∈ SO(n). Exercise 2.1.4. Prove these statements. The space of functionals satisfying (1), (2), (3) is denoted by Val = Val(Rn ). The subspace satisfying in addition (4) is denoted ValSO(n) . Theorem 2.1.5 (Hadwiger [45]).   ValSO(n) = μ0 , . . . , μn . Since μk (Br ) = ck rk , it is clear that the μi are linearly independent —in fact given any n + 1 distinct radii 0 ≤ r0 < · · · < rn ,  det μi (Brj ) ij = 0.

(2.1.28)

Indeed, (2.1.28) is (up to a constant) the determinant of a Vandermonde matrix. Hadwiger’s Theorem yields immediately the following key fact. Let SO(n) := SO(n)  Rn denote the group of orientation-preserving Euclidean motions, with Haar measure d¯ g . The natural normalization is the product of the Lebesgue measure vol on Rn with the probability measure on SO(n), i.e., we stipulate that   d¯ g g¯ : g¯(0) ∈ E = vol(E)

(2.1.29)

for every measurable E ⊂ Rn . Theorem 2.1.6 (Blaschke kinematic formulas). There exist constants ckij such that 

  g= μk A ∩ g¯B d¯ SO(n)



ckij μi (A)μj (B)

(2.1.30)

i+j=n+k

for all A, B ∈ K. Theorem 2.1.7 (Additive kinematic formulas). There exist constants dkij , i+j = k, such that     μk A + gB dg = dkij μi (A)μj (B) (2.1.31) SO(n)

i+j=k

for all A, B ∈ K. Proof of Theorems 2.1.6 and 2.1.7. Consider first (2.1.30). Fixing B, it is clear that the left-hand sides of (2.1.30) and (2.1.31) are SO(n)-invariant valuations in the variable A ∈ K. Furthermore, if Ai → A, then Ai ∩ g¯B → A ∩ g¯B for fixed g¯ ∈ SO(n) provided either A and g¯B are disjoint, or the interiors of A and g¯B intersect. Thus μk (Ai ∩ g¯B) → μk (A ∩ g¯B) for such g¯, which clearly constitute a subset of SO(n) of full measure. Finally, all μk (Ai ∩ g¯B) ≤ μk (B). Thus the

56

Chapter 2. Algebraic Integral Geometry

dominated convergence theorem implies that the left-hand side of (2.1.30) is in ValSO(n) , and therefore  μk (A ∩ g¯B) d¯ g= SO(n)

n 

ci (B)μi (A)

(2.1.32)

i=0

for some constants ci (B). In view of (2.1.28), there are A0 , . . . , An ∈ K and constants αji such that n  j=0



  μk Aj ∩ g¯B d¯ g = ci (B),

αji

i = 0, . . . , n.

(2.1.33)

SO(n)

Repeating the argument of the first paragraph, this time with the Ai fixed and B as the variable, it follows that ci ∈ ValSO(n) for all i, and Hadwiger’s Theorem implies that the right-hand side can be expanded as indicated —simple considerations of scaling ensure that the coefficients vanish unless i + j = n + k. The proof of Theorem 2.1.7 is completely similar.  These facts may be encoded by defining the kinematic and additive kinematic operators kSO(n) , aSO(n) : ValSO(n) → ValSO(n) ⊗ ValSO(n) by kSO(n) (μk ) =



ckij μi ⊗ μj ,

i+j=n+k

aSO(n) (μk ) =



dkij μi ⊗ μj ,

i+j=k

where the ckij and dkij are taken from (2.1.30) and (2.1.31), respectively. Note that if we use the invariant probability measure on SO(n), then kSO(n) (χ) = aSO(n) (vol). Observe that although the kinematic formulas involve curvature measures in general, they give rise to the first-order formulas    g = cn−k−l voln−k−l M k ∩ g¯N l d¯ volk (M k ) voll (N l ) (2.1.34) kl SO(n)

for compact C 1 submanifolds M k and N l (or more generally for suitably rectifiable sets of these dimensions). These are often called Poincar´e–Crofton formulas, but we will use this different term because of the frequent appearance of these names in other parts of the theory. The classical approach to determining the structure constants ckij and dkij is the template method : find enough bodies A and B for which the kinematic integral can be computed directly, and solve the resulting system of linear equations. This

2.1. Classical integral geometry

57

is a bit tricky in the case of Theorem 2.1.6, but for Theorem 2.1.7 it is easy. Let us rescale the μi by ψi = μi (B1 )−1 μi so that ψi (Br ) = ri . Let A = Br and B = Bs be balls of radii r and s, and take dg to be the probability Haar measure on SO(n). Then         k (r + s) = ψk Br + gBs dg = dkij ψi Br ψj Bs SO(n)

i+j=k

=



dkij ri sj ,

i+j=k

so

 k    aSO(n) ψk = ψi ⊗ ψ j . i i+j=k

Exercise 2.1.8. Show that the operators kSO(n) and aSO(n) are coassociative, cocommutative coproducts on ValSO(n) . (Recall that the product on an algebra A is a map A ⊗ A → A, and that the associativity and commutativity properties may be stated in terms of commutative diagrams involving the product map. Thus a coproduct is a map A → A ⊗ A, and the coassociativity and cocommutativity properties are those obtained by reversing all the arrows in these diagrams.) Theorem 2.1.9 (Nijenhuis [52]). There is a basis θ0 , . . . , θn for ValSO(n) such that   θi ⊗ θj and aSO(n) (θk ) = θi ⊗ θj , kSO(n) (θk ) = i+j=n+k

i+j=k

i.e., the structure constants for both coproducts are identically equal to unity. Start of proof. Put θk := ψk /k!. Clearly the second relation holds with θi = θi , but we will see below that the first relation does not —there is a positive constant i c = 1 in front of the right-hand side. But it is easy to check that θi := c n θi works.  Nijenhuis speculated that some algebraic interpretation of the kinematic formulas should explain Theorem 2.1.9. We will see that this is indeed the case.

2.1.5

The Weyl principle

H. Weyl discovered that the Federer curvature measures of a smoothly embedded submanifold of Euclidean space are integrals of Riemannian invariants now commonly known as the Lipschitz–Killing curvatures. In particular, the Federer curvature measures of a smooth submanifold depend only on its intrinsic metric structure and not on the choice of embedding in Euclidean space. This is even

58

Chapter 2. Algebraic Integral Geometry

true for manifolds with boundary. In fact, the Weyl principle applies much more broadly, but we have no systematic understanding of this phenomenon. Let M k ⊂ Rn be a smooth compact submanifold, and let e1 , . . . , ek be a local orthonormal frame, with associated coframe θi and curvature 2-forms Ωi,j . Theorem 2.1.10 ([63]).  ⎧ 1 (k−i) ⎪ 2 ⎪ ⎨(2/π)

1  σa Ωa1 ,a2 · · · Ωa2i−1 ,a2i θa2i+1 · · · θak , M k! μi (M k ) = 0 ≤ k − i, even, ⎪ ⎪ ⎩ 0, otherwise, (2.1.35) where the sum is over all permutations a of 1, . . . , k and σa is the sign. Proof. Put l := n−k. Let ek+1 , . . . , en be a local orthonormal frame for the normal bundle of M . We extend the total frame e1 , . . . , en to a small tubular neighborhood of M by taking ei := ei ◦ πM . Thus ek+1 , . . . , en are parallel to the fibers of πM . Then the tube of radius r around M may be expressed locally as the image of M × Br under the map φ : M × Rl −→ Rn ,

l    φ p, y := p + yi ek+i (p). i=1

Put θi for the coframe dual to ei , with corresponding connection forms ωi,j = ej · dei . The structure equations are dθi =



ωi,j θj ,

(2.1.36)

j

dωi,j =



ωi,r ωr,j = −

r



ωi,r ωj,r ,

(2.1.37)

r

and if i, j ≤ k, dωi,j = Ωi,j −



ωi,s ωj,s ,

(2.1.38)

s≤k

where Ωi,j are the curvature forms. Thus Ωi,j = −



ωi,t ωj,t .

(2.1.39)

t>k

To compute the volume of the tube we note that  ∗

φ θi =

θi +



dyi−k ,

j

yj ωk+j,i ,

if i ≤ k; if i > k.

(2.1.40)

2.1. Classical integral geometry

59

Hence   φ∗ d vol = φ∗ θ1 · · · θn       yj ωk+j,1 · · · θk + yj ωk+j,k dy1 · · · dyl . = θ1 + j

(2.1.41)

j

We wish to integrate this over sets U × B(0, r), where U ⊂ M is open. By symmetry it is clear that any term of odd degree in any of the yi will integrate to 0. Furthermore, the even terms are given by certain products of the θi and ωi,j with coefficients c(e1 , . . . , el ), where    (e1 − 1)!! · · · (el − 1)!! c e1 , . . . , el := , y1e1 · · · ylel dy1 · · · dyl = ωl rl+e (l + 2)(l + 4) · · · (l + e) B(0,r) (2.1.42)

where the ei are

even and e := ei , using the standard trick of multiplying by the Gaussian exp(− yj2 ) and integrating over the space of all y by iteration (cf. [63]). Consider the terms of degree e in y . These contribute to the coefficient of rl+e in the volume of the tube. For definiteness, we consider the terms that are multiples of θe+1 · · · θk dy1 · · · dyl . The remaining factor is        (2.1.43) (−1)π c e1 , . . . , el ωk+1,i · · · ωk+l,i π

i∈π1

i∈πl

over ordered partitions π of {1, . . . , e} into subsets of cardinality e1 , . . . , el . We claim that this sum is precisely ωl rl+e /((l + 2)(l + 4) · · · (l + e)) times the Pfaffian    (−1)π Ωπ1 ,π2 · · · Ωπe−1 ,πe (2.1.44) Pf Ωa,b 1≤a,b≤e = π

of the antisymmetric matrix of 2-forms [Ωa,b ]1≤a,b≤e . Here π ranges over all unordered partitions of {1, . . . , e} into pairs. Observe that there are precisely (e − 1)!! =

e! 2 ( 2e )! e 2

terms in this sum. To prove our claim, we expand each term in the Pfaffian in terms of the ωi,j using (2.1.39). For example,     e Ω1,2 Ω3,4 · · · Ωe−1,e = (−1) 2 ω1,t ω2,t · · · ωe−1,t ωe,t t>k

= (−1)

e 2

  π

i∈π1



ωk+1,i · · ·

t>k



 ωk+l,i ,

i∈πl

where now the sum is over ordered partitions of {1, . . . , e} into l subsets that group each pair 2i − 1, 2i together. Thus the expansion of the Pfaffian yields a

60

Chapter 2. Algebraic Integral Geometry

sum involving precisely the same terms as in (2.1.43), where the coefficients c are replaced by d(π) := the number of partitions, subordinate to π, into pairs. As above, this is precisely d(π) = (e1 − 1)!! · · · (el − 1)!!, which establishes our claim. The coefficient of the integral in (2.1.35) arises from the identity ωn−k = ωn−i (n − k + 2)(n − k + 4) · · · (n − i)

  k−i 2 2 π

for i ≤ k and k − i even.



There are also intrinsic formulas [31] for μi (D) for smooth compact domains D ⊂ M , involving integrals of invariants of the second fundamental form of the boundary of D relative to M . Thus Theorem 2.1.11. If φ : Rn ⊃ M k → RN is a smooth isometric embedding and D ⊂ M is a smooth compact domain, then μi (D) = μi (φ(D)).

2.2

Curvature measures and the normal cycle

2.2.1 Properties of the normal cycle The definition (2.1.10) of the normal cycle N (A) extends to any A ∈ K, regardless of smoothness. As we have seen, it is a naturally oriented Lipschitz submanifold of dimension n − 1, without boundary, in the sphere bundle SRn . It will sometimes be useful to think of N (A) as acting directly on differential forms by integration, i.e., as a current. In these terms, Stokes’ theorem implies that N (A) annihilates all exact forms. The normal

ncycle is also Legendrian, i.e., annihilates all multiples of the contact form α = i=1 vi dxi . This is clear for N (Ar ), since ∂Ar is a C 1 hypersurface and the normal vector annihilates the tangent spaces. It then follows for N (A), since N (Ar ) → N (A) as r ↓ 0. Here the convergence is in the sense of the flat metric: the difference N (Ar ) − N (A) equals ∂Tr , where Tr is the n-dimensional Lipschitz manifold    Tr := x, ∇x δA : 0 < δA (x) < r . Clearly, voln (Tr ) → 0, which is the substance of the flat convergence. By Stokes’ theorem this entails weak convergence. In fact, the operator N is itself a continuous current-valued valuation. Theorem 2.2.1. If K  Ai → A in the Hausdorff metric, then N (Ai ) → N (A) in the flat metric. If A, B, A ∪ B ∈ K, then     N A ∪ B + N A ∩ B = N (A) + N (B). (2.2.1)

2.2. Curvature measures and the normal cycle

61

Proof. The first assertion follows at once from the argument above, together with the observation that for fixed r > 0 the normal cycle N (Ar ) is a small perturbation (with respect to the flat metric) of N ((Ai )r ) for large i. The second assertion is very plausible pictorially, and may admit a nice simple proof. However we will give a different sort of proof based on a larger principle. To begin, note that for generic v ∈ S n−1 the halfspace Hv,c := {x : x · v ≤ c} ⊂ Rn meets A if and only if there is a unique point x ∈ A∩Hv,c such that (x, −v) ∈ N (A), and in fact the intersection multiplicity (Hv,c × {−v}) · N (A) is exactly +1. In other words,     (2.2.2) Hv,c × {−v} · N (A) = χ A ∩ Hv,c for generic v and all c. This condition is enough to determine the compactly supported Legendrian cycle N (A) uniquely [36]. Thus     Hv,c × {−v} · N (A) + N (B) − N (A ∩ B)          = χ A ∩ Hv,c + χ B ∩ Hv,c − χ A ∩ B ∩ Hv,c = χ A ∪ B ∩ Hv,c by the additivity of the Euler characteristic. Hence the uniqueness statement above ensures (2.2.1).  This argument also shows that the normal cycle of a finite union of compact convex sets set is well defined, i.e., the inclusion-exclusion principle yields the same answer regardless of the chosen decomposition into convex sets. The theorem of [36] also implies the existence and uniqueness of normal cycles for much more exotic sets, but this has to do with the analytic side of the subject, which we do not discuss here.

2.2.2 General curvature measures It follows from the valuation property (2.2.1) that any smooth differential form ϕ ∈ Ωn−1 (SRn ) gives rise to a continuous valuation Ψϕ on K. If ϕ is translationinvariant, then so is Ψϕ . The map from differential forms to valuations factors through the space of curvature measures, as follows. Given A ∈ K, put ΦA ϕ for the signed measure  ϕ. ΦA ϕ (E) := N (A)∩πR−1 nE

Denote the space of all such assignments by Curv(Rn ). As in the case of the Federer curvature measures (where ϕ = κi , i = 0, . . . , n − 1), if A ∈ Ksm , then   A ϕ IIx dx, (2.2.3) Px,n(x) Ψϕ (A) = ∂A

ϕ where for each (x, v) ∈ SR the integrand Px,v is a certain type of polynomial in ⊥ A symmetric bilinear forms on v , and IIx is the second fundamental form of ∂A at x. n

62

Chapter 2. Algebraic Integral Geometry

To be more precise about the polynomial P , fix a point (x, v) ∈ SRn and consider the tangent space Tx,v SRn  Rn ⊕ v ⊥  v ⊥ ⊕ v ⊕ v ⊥ =: Q ⊕ v, where Q = α⊥ is the contact plane at (x, v). Note that the restriction of dα is a symplectic form on Q. Now the polynomial P can be characterized as follows. If A ∈ Ksm and (x, v) ∈ N (A) (i.e., v = nA (x)), put L : v ⊥ → v ⊥ for the Weingarten map of ∂A at x. Then graph L ⊂ Q equals the tangent space to N (A) at (x, v) and is a Lagrangian subspace of Q. This is equivalent to the well-known fact ¯ : v ⊥ → Q for the graphing map that the Weingarten map is self-adjoint. Put L ¯ ¯ ∗ ϕx,v as a differential form L(z) := (z, Lz). Then the integrand of (2.2.3) is L on ∂A. Lemma 2.2.2. Let V be a Euclidean space of dimension m and ϕ ∈ Λm (V ⊕ V ). ¯ ∗ ϕ = 0 for all self-adjoint linear maps L : V → V if and only if ϕ is a Then L multiple of the natural symplectic form on V ⊕ V . Proof. The self-adjoint condition on L is equivalent to the condition that the graph ¯ ∗ ϕ = 0 implies that of L be Lagrangian, so it is only necessary to prove that L 2 ∗  ϕ ∈ (ω), where ω ∈ Λ (V ⊕ V ) is the symplectic form. Proposition 2.2.3. The curvature measure Φϕ is 0 if and only if ϕ ∈ (α, dα). Proof. This follows at once from Lemma 2.2.2 and the preceding discussion.



In Theorem 2.4.7 below, due to Bernig and Br¨ ocker [23], we will characterize the kernel of the full map Ψ : Ωn−1 (S ∗ W )W → Val(W ).

2.2.3 Kinematic formulas for invariant curvature measures Let M be a connected Riemannian manifold of dimension n and G a Lie group acting effectively and isotropically on M , i.e., acting by isometries and so that the induced action on the tangent sphere bundle SM is transitive. Under these conditions, the group G may be identified with a sub-bundle of the bundle of orthonormal frames on M . The main examples are the (real) space forms M with their groups G of orientation-preserving isometries; the complex space forms CP n and CH n with their groups of holomorphic isometries, and M = Cn with G = U (n); and the quaternionic space forms. Let K ⊂ H ⊂ G be the subgroups fixing points o¯ ∈ SM , o = π(¯ o) ∈ M . Put CurvG (M )  Λn−1 (To¯SM )K /(α, dα) for the space of G-invariant curvature measures on M . Theorem 2.2.4 ([36]). There is a linear map k˜ = k˜G : CurvG (M ) −→ CurvG (M ) ⊗ CurvG (M )

2.2. Curvature measures and the normal cycle

63

such that, for any open sets U , V ⊂ M and any sufficiently nice compact sets A, B ⊂ M,      ˜ ϕ A ∩ gB, U ∩ gV dg. (2.2.4) k(ϕ) A, U ; B, V = G

Proof. Consider the cartesian square of (G × G)-spaces E

/ G × SM

 SM × SM

 / M × M,

where the vertical bundles have fiber H × So M . Here the map on the right is (g, ξ) → (gπξ, πξ) and the action of G×G on G×SM is (h, k)·(g, ξ) := (hgk −1 , kξ). The fiber of the map on the left over a point (η, ζ) is    F(η,ζ) := g, ξ : g −1 πη = πζ = πξ ⊂ G × SM. Put C(η,ζ) := clos



  g, ξ : ξ = ag −1 η + bζ for some a, b > 0 ⊂ F(η,ζ) .

(2.2.5)

In other words, C(η,ζ) is a stratified space of dimension dim H + 1 and consisting of all pairs (g, ξ) such that g −1 η and ζ lie in a common tangent space, and ξ lies on some geodesic joining these points in the sphere of this tangent space. One may check directly that (h, k) · C(η,ζ) = C(hη,kζ) for (h, k) ∈ G × G. Each C(η,ζ) carries a natural orientation such that        ∂C(η,ζ) = H(η,ζ) × {ζ} − g, g −1 η : g ∈ H(η,ζ) + K(η,ζ) × Sπζ M , where H(η,ζ) := {g ∈ G : πg −1 η = πζ} and H(η,ζ) ⊃ K(η,ζ) := {g : g −1 η = −ζ}. For β ∈ Ω∗ (SM )G we have    G×G , πC∗ dg ∧ β ∈ Ω∗ (SM ) × Ω∗ (SM ) where πC∗ is fiber integration over C. Now consider C(A, B) := N (A)×N (B)×E C ⊂ E. One checks that the image of C(A, B) in G × SM consists of all (g, ξ) such that g −1 πη = πζ = πξ for some η ∈ N (A), ζ ∈ N (B) and ξ lying on a geodesic between g −1 η and ζ. Furthermore, the set of those g for which g −1 η = −ζ for some η ∈ N (A) and ζ ∈ N (B) has codimension 1: it is the image in G of N (A) × N (B) ×E K, which has dimension 2n − 2 + dim K, whereas dim G = dim G/K + dim K = dim SM + dim K = 2n − 1 + dim K. Thus A and gB meet transversely for a.e. g ∈ G, and for such g we have     −1 (g) . N A ∩ gB = N (A)π −1 (gB) + g∗ N (B)π −1 A + πSM ∗ C(A, B) ∩ πG

64

Chapter 2. Algebraic Integral Geometry

Now we may compute the kinematic integral for a given β ∈ Ωn−1 (SM )G in either of two ways: either by pushing N (A)×N (B)×E C into the top right corner G×SM of (2.2.3) and integrating dg ∧ β; or else by pulling back dg ∧ β to E, pushing it down to SM × SM via πC∗ , and integrating the result over N (A) × N (B). Thus the conclusion of the theorem is fulfilled for the curvature measure ϕ = Φβ with   ˜ β ) = Φβ ⊗ vol + vol ⊗Φβ + Φ ⊗ Φ , k(Φ π (dg∧β) C∗

where volA (U ) := vol(A ∩ U ).

2.2.4



The transfer principle

The following is an instance of the general transfer principle of Howard [46]. Let M± be two Riemannian manifolds of dimension n, and G± be Lie groups acting isotropically on M± . Assume further that the subgroups H± fixing chosen points o± ∈ M± are isomorphic, and that there is an isometry ι : To+ M+ −→ To− M−

(2.2.6)

intertwining the actions of H± . We identify H± with a common model H. Let K ⊂ o± ) = o± , H be the subgroup of points fixing a chosen point o¯± ∈ SM± with π± (¯ where π± : SM± → M± is the projection. Since the actions of G± are isotropic, we may assume that ι(¯ o+ ) = o¯− . To simplify notation we denote the points o± and o¯± by o and o¯, respectively. The tangent spaces to the sphere bundles may be decomposed into the horizontal (with respect to the Riemannian connection) and vertical subspaces: To¯SM± = P± ⊕ V± Thus there are canonical isomorphisms P±  T o M ± ,

V±  o¯⊥ ⊂ To M±

and therefore ι induces a K-equivariant isomorphism ¯ι : To SM+ → To SM− . This gives an isomorphism of exterior algebras  G−  K  K  G+  Λ∗ To¯SM− −→ Λ∗ To¯SM+  Ω∗ SM+ . ¯ι∗ : Ω∗ SM− Since the contact form α restricts to the inner product with o¯ on the P factor, and to 0 on V , it follows that ¯ι∗ α = α. Although ¯ι∗ is not an isomorphism of differential algebras (i.e., does not intertwine d), nevertheless ¯ι∗ dα = dα. This

n−1 can be seen directly since in each case dα = i=1 θi ∧ θ˜i , where θi and θ˜i are orthonormal coframes for o¯⊥ ⊂ P and for V that correspond under the natural isomorphism —in other words, dα corresponds to the natural symplectic form on the cotangent bundles. Therefore, ¯ι∗ induces a natural isomorphism  ι : CurvG− (M− ) −→ CurvG+ (M+ )

(2.2.7)

2.2. Curvature measures and the normal cycle

65

via the identifications CurvG± (M± )  Ωn−1 (SM± )G± /(α, dα). Theorem 2.2.5. If there exists an isometry (2.2.6) as above, then the following diagram commutes:  ι

CurvG− (M− )  k G−

 CurvG− (M− ) ⊗ CurvG− (M− )

/ CurvG+ (M+ )

(2.2.8)

 k G−

 ι⊗ ι

 / CurvG+ (M+ ) ⊗ CurvG+ (M+ ).

Proof. Note that each G± may be identified with a subbundle G± of the orthonormal frame bundle F± of M± : select an arbitrary orthonormal frame f at some point o ∈ M± , and take G± to be the G± orbit of f . Since G± acts effectively, the induced map is a diffeomorphism. Recall [27] that the Riemannian metric on M± induces a canonical invariant horizontal distribution D on F± , with each plane Df ⊂ Tf F± linearly isomorphic to Tπf M± , where π : F± → M± is the projection. A modification of the Sasaki metric endows F± with a Riemannian structure g, where each Df ⊥ π −1 πf ; the restriction of g to each fiber π −1 x is induced by the standard invariant metric on SO(n)  π −1 x ⊂ F± . Also, the restriction of π∗ to Df is an isometry to Tπf M . For f ∈ G± consider the orthogonal projection, with respect to this Sasaki metric, of Df to Tf G± ⊂ Tf F± . This yields a distribution M± on G± , complementary to the tangent spaces of the fibers of G± , and clearly invariant under the action of G± (since this action is by isometries). Putting m± := M±,o ⊂ To G±  Te G± = g± , it follows that at the level of Lie algebras there is a natural decomposition (2.2.9) g± = h ⊕ m± , where m± is naturally identified with To M± under the projection map, and is invariant under the adjoint action of H (i.e., the homogeneous space G/H is reductive). Furthermore, the isometry ι induces an H-equivariant isomorphism m+ → m− . We claim that the maps k˜G± depend only on the data above. To see this, for simplicity of notation we drop the ±, and consider the diagram of derivatives of (2.2.3) over the point (¯ o, o¯) ∈ SM × SM for the pairs (M± , G± ): T E|F¯

/ T (G × SM )| F

 To¯SM × To¯SM

 / To M × To M,

(2.2.10)

where F = H × H/K and F¯ are the fibers over (o, o) and (¯ o, o¯), respectively. The vertical bundles have fiber T H × T (So M ) = T H × T (H/K), and the diagram is

66

Chapter 2. Algebraic Integral Geometry

again a cartesian square. Writing this in terms of the Lie algebras, we have T E|F¯

/ T H ⊕ m ⊕ T (H/K) ⊕ m

 h/k ⊕ m ⊕ h/k ⊕ m

 / m⊕m

(2.2.11)

since T G|H = T H ⊕ m. The maps on the bottom and on the right are the obvious projections. These diagrams are (H × H)-equivariant, and the H × H actions depend only on the structure of m as an H-module. In particular, we may re-insert the ± and the diagram obtained from the obvious maps between the two diagrams is commutative and (H × H)-equivariant. The restrictions of the G-invariant forms on SM are precisely the H-invariant sections of the exterior algebra bundles of the various bundles occurring here. Our convention on volume forms dictates that the distinguished volume form of G restricts to the product of the probability volume form on H with the natural volume form on m arising from the identification with To M . In other words, the natural maps between the ± diagrams respect the mapping ¯ι. Finally, it is clear that the spaces (or currents) C from (2.2.5) also correspond under these maps, hence the maps between the ± spaces also intertwine the fiber integrals over C, which is the assertion of the theorem.  Theorem 2.2.5 implies, for example, that in some sense the kinematic formulas of all three space forms Rn , S n , and H n are the same, in the sense that there is a canonical identification of the spaces of invariant curvature measures on S n (or H n ) and Rn which moreover intertwines the respective kinematic opera˜ these spaces are the quotients Gλ /SO(n) for λ = 0, 1, −1, respectively, tors k: where G0 = SO(n) × Rn , G1 = SO(n + 1) and G−1 = SO(n, 1). We take H = SO(n) ⊃ K := SO(n − 1). Their Lie algebras are the subalgebras ⎧⎡ 0 ⎪ ⎪ ⎪ ⎨⎢ a10 ⎢ gλ = ⎢ . ⎪ ⎣ .. ⎪ ⎪ ⎩ an0

a01

··· h

a0n

⎤ ⎥ ⎥ ⎥ : h ∈ so(n), a0i + λai0 ⎦

⎫ ⎪ ⎪ ⎪ ⎬ =0 ⎪ ⎪ ⎪ ⎭

(2.2.12)

of gl(n + 1). The subspace m is {h = 0}. The complex space forms Cn (with the restricted isometry group U (n)), CP n , and CH n admit a similar description, so again it follows that the integral geometry of these spaces at the level of curvature measures is independent of the ambient curvature. However, the case of the real space forms is uniquely simple, due to the fact that the map from SO(n)-invariant curvature measures to valuations is an isomorphism.

2.3. Integral geometry of Euclidean spaces via Alesker theory

2.3

67

Integral geometry of Euclidean spaces via Alesker theory

2.3.1 Survey of valuations on finite-dimensional real vector spaces The recent work of S. Alesker revolves around a deepened understanding of convex valuations. Given a finite-dimensional real vector space W of dimension n, consider the space Val = Val(W ) of continuous translation-invariant convex valuations on W . For convenience we will assume that W is endowed with a Euclidean structure, although this device may be removed by inserting the dual space W ∗ appropriately. The valuation ϕ ∈ Val(W ) is said to have degree k and parity  = ±1 if ϕ(tK) = tk ϕ(K)

(t > 0),

ϕ(−K) = ϕ(K), for all K ∈ K(W ). Denote the subspace of valuations of degree k and parity  on W by Valk, (W ) ⊂ Val(W ). Putting   ϕ := sup ϕ(K) : K ∈ K(W ), K ⊂ B1 , where B1 is the closed unit ball in W , gives Val(W ) the structure of a Banach space. The group GL(W ) acts on Val(W ) by g · ϕ(K) := ϕ(g −1 K), and this action stabilizes each Valk, . Put Valsm (W ) for the subspace of smooth valuations, i.e., the space of valuations ϕ such that the map GL(W ) → Val, g → gϕ, is smooth. General theory (cf. [2]) ensures that Valsm is dense in Val. The starting point for Alesker’s approach is Theorem 2.3.1 (Irreducibility Theorem [2, 51]). As a GL(W )-module, the decomposition of Val(W ) into irreducible components is  Valk, (W ). (2.3.1) Val(W ) = k=0,...,n; =±1

Furthermore, Val0 and Valn are both 1-dimensional, spanned by the Euler characteristic χ and the volume vol, respectively. Irreducibility here means that each Valk, (W ) admits no nontrivial, closed, GL(W )-invariant subspace. For A ∈ K(W ) we define the valuation μA ∈ Val(W ) by      μA (K) := vol A + K = χ (x − A) ∩ K dx. (2.3.2) W

The integrand of course takes only the values 0 and 1, depending on whether or not the intersection is empty.

68

Chapter 2. Algebraic Integral Geometry

Given an even valuation ϕ of degree k we say that a signed measure mϕ on Grk (W ) is a Crofton measure for ϕ if    ϕ(A) = volk πE (A) dmϕ (E). Grk

If k = 1 or n − 1, then mϕ is uniquely determined by ϕ, but not in the remaining cases 2 ≤ k ≤ n − 2. Proposition 2.1.2 means that the Haar measure on Grk is a Crofton measure for μk , k = 0, . . . , n. Klain [48] proved that an even valuation ϕ ∈ Valk,+ is uniquely determined by its Klain function Klϕ : Grk −→ R, which is defined by the condition that the restriction of ϕ to E ∈ Grk is equal to Klϕ (E) volk |E . Our next statement summarizes the most important implications of Theorem 2.3.1 for integral geometry. Theorem 2.3.2 (Alesker [3, 4, 5, 6, 11]). (1) The following equality holds: 

n     Valsm voln ⊕ Ψγ : γ ∈ Ωn−1 (SW )W = k (W ).

(2.3.3)

k=0

(2) Every smooth even valuation of degree k admits a smooth Crofton measure. (3) μA : A ∈ K(W ) is dense in Val(W ). (4) There is a natural continuous commutative graded product on Valsm (W ) such that      μA · ϕ (K) = ϕ (x − A) ∩ K dx. (2.3.4) W

In particular, the multiplicative identity is the Euler characteristic χ. (5) (Alesker Fourier transform). There is a natural linear isomorphism sm &: Valsm k, −→ Valn−k,

such that (' ϕ) ˆ = ϕ. For smooth even valuations, this map is given in terms of Crofton measures by (2.3.5) mϕˆ =⊥∗ mϕ , where ⊥ : Grk → Grn−k is the orthogonal complement map. Equivalently, Klϕˆ = Klϕ ◦ ⊥ .

(2.3.6)

(6) The product satisfies Poincar´e duality, in the sense that the pairing (ϕ, ψ) := degree n part of ϕ · ψ ∈ Valn  R is perfect. The Poincar´e pairing is invariant under the Fourier transform:   ϕ, ˆ ψˆ = (ϕ, ψ). (2.3.7)

2.3. Integral geometry of Euclidean spaces via Alesker theory

69

(7) (Hard Lefschetz). The degree one map L ϕ := μ1 · ϕ satisfies the hard Lefschetz property: for k ≤ 12 n the map sm (L )n−2k : Valsm k −→ Valn−k

is a linear isomorphism. (8) If G ⊂ SO(W ) acts transitively on the sphere of W , then the subspace of G-invariant valuations has dim ValG (W ) < ∞. Furthermore, ValG (W ) ⊂ Valsm (W ). Remark 2.3.3. (1) Theorem 2.3.1 implies that the space on the left-hand side of (2.3.3) is dense in Val(W ); that this space consists precisely of the smooth valuations follows from general representation theory (Casselman–Wallach theorem). (2) The groups G as in (8) above have been classified (cf. Alesker’s lecture notes in this book). In fact it is true that, modulo the volume valuation, every ϕ ∈ ValG is given by integration against the normal cycle of some G-invariant form on the sphere bundle. This implies immediately that dim ValG < ∞. Intertwining the Fourier transform and the product we obtain the convolution on Valsm :  ˆ ϕ ∗ ψ := ϕ ˆ · ψ. (2.3.8) Recall that if A1 , . . . , An−k ∈ K then ∂ n−k k! μ ti Ai (B) V A1 , . . . , An−k , B[k] := n! ∂t1 · · · ∂tn−k t1 =···=tn−k =0 



is the associated mixed volume, which is a translation-invariant valuation of degree k in B. Here B[k] denotes the k-tuple (B, . . . , B). Theorem 2.3.4 ([11, 24]). (2.3.9) μA ∗ ϕ = ϕ( · + A),  k! l!   V A1 , . . . , An−k , B1 , . . . , Bn−l , · . V A1 , . . . , An−k , · ∗ V B1 , . . . , Bn−l , · = n! (2.3.10) 





Exercise 2.3.5. Prove that (2.3.10) follows from (2.3.9). Corollary 2.3.6. If we define the degree −1 operator Λ by ϕ),  Λ ϕ := (L ˆ

then

1 d Λ ϕ = μn−1 ∗ ϕ = ϕ( · + Br ). 2 dr r=0

(2.3.11)

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Chapter 2. Algebraic Integral Geometry

This may in turn be expressed in terms of the underlying differential forms as Λ Ψ θ =

1 ΨL θ , 2 T

where T (x, v) := (v, 0) is the Reeb vector field of T Rn and L is the Lie derivative. It is convenient to renormalize the operators L and Λ by putting 2ωk L, ωk+1 2ωn−k Λ Λ := ωn−k+1 L :=

(2.3.12) (2.3.13)

on valuations of degree k. Thus  Λϕ = (L ϕ). ˆ

(2.3.14)

In view of relation (2.3.27) below, these act on the intrinsic volumes as Lμk = (k + 1)μk+1 ,

(2.3.15)

Λμk = (n − k + 1)μk−1 .

(2.3.16)

2.3.2 Constant coefficient valuations The normal cycle of a smooth submanifold is its unit normal bundle, so it makes sense to think of N (A) for general sets A as a generalization of this concept. It is also convenient to consider the analogue N1 (A) ⊂ T Rn  Rn × Rn of the bundle of unit normal balls, obtained by summing A × {0} and the image of N (A) × [0, 1] by the homothety map in the second factor. Thus ∂N1 (A) = N (A). If a differential form ϕ ∈ Ωn−1 (SRn ) extends to a smooth form on all of Rn ⊕ Rn , then Stokes’ theorem yields   ϕ= dϕ. Ψϕ := N( · )

N1 ( · )

If dϕ happens to have constant coefficients (i.e., is invariant under translations of both the base and the fiber), then a number of simplifications ensue. Definition 2.3.7. μ ∈ Valsm (V ) is a constant coefficient valuation if there exists θ ∈ Λn (V ⊕ V ) such that  μ = Ψθ := θ. N1 ( · )

We denote by CCV ⊂ Val(V ) the finite-dimensional vector subspace consisting of all constant coefficient valuations.

2.3. Integral geometry of Euclidean spaces via Alesker theory

71

Exercise 2.3.8. Show that every constant coefficient valuation is smooth and even. This concept only makes sense with respect to the given Euclidean structure on V —the spaces of constant coefficient valuations associated to different Euclidean structures are different. Using Stokes’ theorem it is easy to see that ValSO(n) ⊂ CCV , i.e., that the intrinsic volumes μi are all constant coefficient valuations. We will see below that, if V = Cn , then this is also true of all valuations invariant under U (n). When restricted to constant coefficient valuations, the actions of L, Λ, and the Fourier transform admit the following simple algebraic model. Consider adapted coordinates x1 , . . . , xn , y1 , . . . , yn

n n for R ⊕ R , so that ω = dxi ∧ dyi is the usual symplectic form. Put j : Λ∗ (Rn ⊕ Rn ) −→ Λ∗ (Rn ⊕ Rn ) for the algebra isomorphism that interchanges the coordinates: j(dxi ) = dyi ,

j(dyi ) = dxi ,

i = 1, . . . , n.

There are derivations , λ : Λ∗ (Rn ⊕Rn ) → Λ∗ (Rn ⊕Rn ) of degrees ±1, respectively, determined by (dyi ) := dxi , λ(dxi ) := dyi ,

(dxi ) := 0,

λ(dyi ) := 0, j ◦  = λ ◦ j.

In fact, putting mt (x, y) := (x + ty, y), we have d λϕ = m∗ ϕ dt t=0 t

(2.3.17)

(2.3.18)

and a similar formula holds for . It is easy to see that, for a monomial ϕ ∈ Λ∗ (Rn ⊕ Rn ), [, λ] ϕ = (r − s)ϕ, (2.3.19) where r and s denote the number of dxi and dyi factors in ϕ, respectively. Putting H for the operator ϕ → (r − s) ϕ, clearly [H, ] = 2,

[H, λ] = −2λ.

(2.3.20)

In other words, Lemma 2.3.9. Let X, Y and H, with [X, Y ] = H, [H, X] = 2X and [H, Y ] = −2Y , be generators of sl(2, R). Then the map H −→ r − s,

72

Chapter 2. Algebraic Integral Geometry X −→ , Y −→ λ,

defines a representation of sl(2, R) on Λ∗ (Rn ⊕ Rn ). The subspace Λn (Rn ⊕ Rn ) is naturally graded by the number of factors dxi that appear. Proposition 2.3.10. (1) The surjective map Ψ : Λn (Rn ⊕ Rn ) → CCV is graded. The kernel of Ψ consists precisely of the subspace of multiples of the symplectic form. (2) The operators j, λ and  induce (up to scale) the operators &, Λ and L on CCV , via the formulas in degree k &◦Ψ =

ωn−k Ψ ◦ j, ωk

Λ◦Ψ=

ωn−k Ψ ◦ λ, ωn−k+1

L◦Ψ=

ωn−k Ψ ◦ , ωn−k−1

and satisfying the relation Λ ◦ & = & ◦ L. (3) The map H−

→ 2k − 2n, X−

→ L, Y −

→ Λ, defines a representation of sl(2, R) on CCV . Proof. The first assertion of (1) is obvious, and the second follows at once from Lemma 2.2.2. Noting that j takes the symplectic form to −1 times itself, and that  and λ annihilate it, the first assertion of (2) follows at once. The first formula follows at once from the definition of the Alesker Fourier transform. To prove the second formula, let ϕ ∈ Λn (Rn ⊕ Rn ) and let θ ∈ Ωn−1 (Rn × Rn ) be a primitive. Then, by (2.3.11) and (2.3.18), Λ Ψ ϕ = Λ Ψ θ  =Λ θ N( · )  1 LT θ = 2 N( · )

2.3. Integral geometry of Euclidean spaces via Alesker theory

73

 1 dLT θ 2 N1 ( · )  1 = LT dθ 2 N1 ( · ) 1 = Ψλϕ , 2

=

from which the second formula follows at once. The fourth formula is (2.3.14), and the third follows from that relation and (2.3.17). Assertion (3) follows by direct calculation from (2) and Lemma 2.3.9.  Problem 2.3.11. What is the maximal subspace of Valsm for which assertion (3) holds? (Alesker has shown that it does not hold for the full algebra Valsm .) Exercise 2.3.12. Prove that if μ is a constant coefficient valuation of degree k and P ⊂ Rn is a convex polytope, then    μ(P ) = Klμ F  volk (F )∠(P, F ), (2.3.21) F ∈Pk

where Pk is the k-skeleton of P and ∠(P, F ) is the appropriately normalized exterior angle of P along F . Problem 2.3.13. Characterize the μ ∈ Val+ k satisfying (2.3.21) for every convex polytope P . (Note that every even valuation of degree n − 1 satisfies (2.3.21), but the space of such valuations is infinite-dimensional.) Problem 2.3.14. The algebras ValSO(n) (Rn ) of SO(n)-invariant valuations on Rn and ValU (n) (Cn ) of U (n)-invariant valuations on Cn are subspaces of CCV , as we will see below. Classify the subalgebras of the constant coefficient valuations. Do they constitute an algebra? If not, what algebra do they generate?

2.3.3 The FTAIG for isotropic structures on Euclidean spaces We note two important consequences of Theorem 2.3.2. First, assertion (1) (together with Corollary 2.4.8 below) implies that every ϕ ∈ Valsm (W ) can be applied not only to elements of K(W ), but also to any compact set that admits a normal cycle, e.g., smooth submanifolds and submanifolds with corners. Second, recalling how Hadwiger’s Theorem 2.1.5 implies the existence of the classical kinematic formulas, Theorem 2.3.2 (8) implies the existence of kinematic formulas for Euclidean spaces V under an isotropic group action. For G ⊂ O(V ), put G := G  V for the semidirect product of G with the translation group. Thus (V, G) is isotropic if and only if G acts transitively on the sphere of V . Proposition 2.3.15 ([4, 36]). In this case there are linear maps kG , aG : ValG −→ ValG ⊗ ValG

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Chapter 2. Algebraic Integral Geometry

such that for K, L ∈ K(V ) and ϕ ∈ ValG ,    ϕ K ∩ g¯L d¯ g, kG (ϕ)(K, L) = G   aG (ϕ)(K, L) = ϕ K + gL dg. G

The kinematic operator kG is related to k˜G via the commutative diagram CurvG (V ) Ψ

 kG

/ CurvG (V ) ⊗ CurvG (V )

(2.3.22)

Ψ⊗Ψ



kG

ValG (V )

 / ValG (V ) ⊗ ValG (V ).

Proof. The statements about kG are direct consequences of Theorem 2.2.4. Invoking (8) of Theorem 2.3.2, the proof of the existence of aG (and another proof of the existence of kG ) follows precisely the proof of Theorems 2.1.6 and 2.1.7 above.  The next theorem may rightly be called the Fundamental Theorem of Algebraic Integral Geometry for Euclidean spaces. Before stating it we need to add some precision to statement (6) of Theorem 2.3.2 by specifying the isomorphism Valn (Rn )  R to take Lebesgue voln  1. Theorem 2.3.16. Let p : ValG → ValG∗ denote the Poincar´e duality map from (6) of Theorem 2.3.2, mG : ValG ⊗ ValG → ValG the restricted multiplication map, and m∗G : ValG∗ → ValG∗ ⊗ ValG∗ its adjoint. Then the following diagram commutes: ValG (V ) p



kG

/ ValG (V ) ⊗ ValG (V )

(2.3.23)

p⊗p

ValG∗ (V )

m∗ G

 / ValG∗ (V ) ⊗ ValG∗ (V ).

The same is true if kG and mG are replaced by aG and cG , respectively, where cG is the convolution product. In particular aG = (& ⊗ &) ◦ kG ◦ &.

(2.3.24)

Proof. The dual space ValG∗ is a ValG -module by setting      α · β ∗ , γ := β ∗ , α · γ . With this definition it is clear that p is a map of ValG -modules, and it is easy to check that m∗G is multiplicative in the sense that       m∗G α · β ∗ = α ⊗ χ · m∗G (β) = χ ⊗ α · m∗G (β).

2.3. Integral geometry of Euclidean spaces via Alesker theory

75

On the other hand, by (3) and (8) of Theorem 2.3.2, the valuations    μG := χ · ∩¯ g A d¯ g = kG (χ)( · , A) (2.3.25) A G

span ValG . Thus we may check the multiplicativity of kG by computing    G    kG μG μA · ϕ B ∩ g¯C d¯ · ϕ (B, C) = g A G     ¯ ∩ g¯C dh ¯ d¯ ϕ B ∩ hA g = G G    ¯ C dh ¯ kG (ϕ) B ∩ hA, = G

=



 μG A ⊗ χ · kG (ϕ) (B, C)

by Fubini’s theorem, for ϕ ∈ ValG . It remains to show that (p ⊗ p)(kG (χ)) = m∗G (p(χ)). We may regard these elements of ValG∗ ⊗ ValG∗ as lying in Hom(ValG , ValG∗ ) instead. Notice that in these terms both elements are graded maps in a natural way. Furthermore, (2.3.25) and the multiplicativity of kG imply that (p ⊗ p)(kG (χ)) is an invertible map of ValG -modules, and clearly the same is true for m∗G (p(χ)). Since dim(Val0 ) = 1, it follows that the two must be equal up to scale. To determine the scaling factor we compare (p ⊗ p)(kG (vol)) and m∗G (p(vol)). But using the facts p(vol) = χ∗ , kG (vol) = vol ⊗ vol (the latter follows from the conventional normalization (2.0.1) of the Haar measure dg), it follows that the scaling factor must be 1. Here χ∗ is the dual element evaluating to 1 on χ and annihilating all valuations of positive degree.  In more practical terms, Theorem 2.3.16 may be summarized in the following statements: (1) Let ν1 , . . . , νN and φ1 , . . . , φN be two bases for ValG , and consider the N × N matrix Mij := p(νi ), φj . Then kG (χ) =



M −1



ν ij i

⊗ φj .

(2.3.26)

i,j

In other words kG (χ) = p−1 ∈ Hom(ValG∗ , ValG ). Hence, kG also determines the restricted product mG . (2) kG is multiplicative, in the sense that kG (ϕ · μ) = (ϕ ⊗ χ) · kG (μ). Bernig has observed the following:

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Chapter 2. Algebraic Integral Geometry

Proposition 2.3.17. If G acts isotropically on Rn , then ValG (Rn ) ⊂ Val+ (Rn ). No general explanation for this fact is known. With (2.3.7) this implies that if φi = νˆi above, then the matrix M is symmetric, and hence so is M −1 .

2.3.4 The classical integral geometry of Rn Modulo α, dα, the space of invariant forms Ωn−1 (SRn )SO(n) equals κ0 , . . . , κn−1 , corresponding to the elementary symmetric functions of the principal curvatures of a hypersurface. With Theorem 2.3.2 this implies Hadwiger’s Theorem 2.1.5. Furthermore, Theorem 2.3.18. As an algebra, ValSO(n) (Rn )  R[t]/(tn+1 ), with ti = Proof. Put t := 

 Grn−1





t = Grn−1

(2.3.27)

χ( · ∩P ) dP . Then, by the definition of the Alesker product,  

t · ∩P dP =

2

i!ωi 2i+1 μi = μi . i π αi



(Grn−1 )2





  χ · ∩R dR,

χ · ∩P ∩ Q dQdP = Grn−2

etc. By (2.1.27), it follows that the powers of t are multiples of the corresponding μi , which establishes the first assertion. To determine the coefficients relating the ti and the μi , we apply the transfer principle (Theorem 2.2.5) to the isotropic pairs (Rn , SO(n)) and (S n , SO(n + 1)). Let Ψi = α2i μi and Ψ i its image under the transfer (i.e., if ψi is the image of the corresponding curvature measure under the transfer map, then Ψ i (A) = ψi A (A) for A ⊂ S n ). Then Ψ i (S j ) = 2δij and the kinematic formula for (S n , SO(n + 1)) is      kS n (Ψ c ) S a , S b = Ψ c S a ∩ gS b dg = 2αn δca+b−n SO(n+1)

under our usual convention for the Haar measure on the group. Thus the template method implies that kS n (Ψ c ) =

αn 2



Ψ a ⊗ Ψ b .

a+b=n+c

Now the multiplicativity of kRn , together with Theorem 2.2.5, yields  αn αn (Ψc ⊗ χ) · Ψa ⊗ Ψb = (Ψc ⊗ χ) · kRn (χ) = kRn (Ψc ) = 2 2 a+b=n

 a+b=n+c

Ψa ⊗ Ψb .

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77

It follows that Ψa · Ψb ≡ Ψa+b . It is convenient however to take t := 2Ψ1 ,

(2.3.28)

whence the second relation of (2.3.27) follows. The first then follows from the identity 2n+1 π n , ωn ωn+1 = (n + 1)! 

so the proof is complete.

Adjusting the normalization of the Haar measure dg and unwinding Theorem 2.3.16, it follows that  ta ⊗ tb , c = 0, . . . , n. (2.3.29) kSO(n) (tc ) = a+b=n+c

In fact dg is chosen so that dg({g : go ∈ S}) = examining the leading term t ⊗ t = first assertion of Theorem 2.1.9. c

n

Corollary 2.3.19.

 μi · μj =

2n+1 c αn t

2n+1 αn

voln (S), as may be seen by

⊗ voln on the right. This yields the

 i + j ωi+j μi+j . i ωi ωj

Remark 2.3.20. The relations (2.3.27) may also be expressed as exp(πt) =



ωi μi ,

 μi 1 = . 2−t αi

(2.3.30)

We say that a valuation ϕ ∈ Val(Rn ) is monotone if ϕ(A) ≥ ϕ(B) whenever A ⊃ B, A, B ∈ Kn ; positive if ϕ(A) ≥ 0 for all A ∈ Kn ; and Crofton positive if each of its homogeneous components admits a nonnegative Crofton measure. Such valuations constitute the respective cones CP ⊂ M ⊂ P ⊂ Val(Rn ). Exercise 2.3.21. Show that CP ∩ ValSO(n) = M ∩ ValSO(n) = P ∩ ValSO(n) = μ0 , . . . , μn + . In fact, the algebra of the vector space of SO(n)-invariant valuations on Rn given in Theorem 2.3.4 is a special case of a more general theorem, due to ´ Alvarez–Fernandes and Bernig, about the Holmes–Thompson volumes associated to a smooth Minkowski space. Unfortunately, however, there is no “dual” interpretation as a kinematic formula in this setting. Recall that a Finsler metric on a smooth manifold M n is a smoothly varying family g of smooth norms on the tangent spaces Tx M . Let gx∗ denote the dual

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Chapter 2. Algebraic Integral Geometry

norm on Tx∗ M , and Bx∗ := (gx∗ )−1 [0, 1] ⊂ Tx∗ M the associated field of unit balls. Then for E ⊂ M the Holmes–Thompson volume of M is the measure  n , (2.3.31) HTg (E) := ωn−1  x∈E

∗ Bx

where  is the natural symplectic form on T ∗ M (cf. [62]). A Minkowski space is the special case of a finite-dimensional normed vector space W n with smooth norm F . In this case the Holmes–Thompson volume of a subset S ⊂ W n may be expressed as HTF (S) = ωn−1 voln (S) voln (BF ∗ ), (2.3.32) where voln is the Lebesgue volume of the background Euclidean structure. For a compact smooth submanifold M m ⊂ W it is natural to define HTF m (M ) to be the total Holmes–Thompson volume of M with respect to the Finsler metric on M induced by F . Extending (2.3.32) this may also be expressed as    F −1 volm B( F |T M )∗ d volm x, (2.3.33) HTm (M ) = ωm x

M

where again volm is the m-dimensional volume induced by the Euclidean structure on W . In particular, if E ∈ Grm (W ) and M ⊂ E, then   −1 ∗ volm (M ). (2.3.34) HTF m (M ) = ωm volm (BF ∩ E) Here the polar is taken as a subset of E. Theorem 2.3.22 ([15, 17]). Each HTF m extends uniquely to a smooth even valuation of degree m on W , which we denote by μF m . Furthermore   i + j ωi+j F F μF · μ = μ . (2.3.35) i j ωi ωj i+j i Thus the vector space spanned by the μF i is in fact a subalgebra of Val(W ), isomorphic to R[x]/(xn+1 ). Proof. It is a general fact that for A ∈ K(W ) and E ∈ Grm (W ) (A ∩ E)∗ = πE (A∗ ),

(2.3.36)

where the polar on the left is as a subset of E, and on the right as a subset of W . Observe that if K ∈ K(E ⊥ ), then (n − m)! dm ∗ V (A [m], K[n − m]) = m voln (K + tA∗ ) n! dt t=0 = m! voln−m (K) volm (πE (A∗ )).

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79

Hence if we take A = BF , then by (2.3.34), (2.3.36) and the characterization of the Fourier transform in terms of Klain functions in Theorem 2.3.2 (5), it follows that   μF m := cV BF ∗ [m], · & is a smooth even valuation of degree m extending the mth Holmes–Thompson volume. Now, by (2.3.8) and (2.3.10),  F ' ' F F μF i · μj = μi ∗ μj     = c V (BF ∗ [i], · ∗ V BF ∗ [j], ·) &   = cV BF ∗ [i + j], · & = cμF i+j . To compute the constant we note that μF i = μi if F is Euclidean, and use the relations (2.3.27). 

2.4 Valuations and integral geometry on isotropic manifolds 2.4.1 Brief definition of valuations on manifolds More recently, Alesker [10] has introduced a general theory of valuations on manifolds, exploiting the insight that smooth valuations may be applied to any compact subset admitting a normal cycle. Formally, given a smooth oriented manifold M of dimension n, the space V(M ) of smooth valuations on M consists of functionals on the space of smooth polyhedra (i.e., smooth submanifolds with corners) P ⊂ M   of the form P → P α + N (P ) β, hence may be identified with a quotient of the space Ωn (M ) × Ωn−1 (SM ). Definition 2.4.1. A smooth valuation on M is determined by a pair (θ, ϕ) ∈ Ωn (M ) × Ωn−1 (S ∗ M ) via   Ψθ,ϕ (A) := θ+ ϕ. (2.4.1) A

N (A)

The space of smooth valuations on M is denoted V(M ). If M is a vector space, then it is natural to consider those smooth valuations determined by translation-invariant differential forms ω and ϕ. It follows from Theorem 2.3.2 (1) that the subspace of all such smooth valuations coincides with Valsm (M ) as defined in Section 2.3.1. The space V(M ) carries a natural filtration, compatible with the grading on the subspace of translation-invariant valuations if M is a vector space. Parallel to Theorem 2.3.2 above is the following.

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Theorem 2.4.2 (Alesker [7, 8, 9, 10, 14]). (1) There is a natural continuous commutative filtered product on V(M ) with multiplicative identity given by the Euler characteristic χ. If N ⊂ M is an embedded submanifold, then the restriction map rN : V(M ) → V(N ) is a homomorphism of algebras. If M is a vector space, then the restriction of this product to translation-invariant valuations coincides with the product of (2.3.4). (2) If M is compact, then the product satisfies Poincar´e duality, in the sense that the pairing (ϕ, ψ) := (ϕ · ψ)(M ) is perfect. (3) Suppose that the Lie group G acts transitively on the sphere bundle SM and put V G (M ) for the space of valuations on M invariant under G. Then dim V G (M ) < ∞. The basic idea of the product is that if X ⊂ M is a “nice” subset —say, a piecewise smooth domain— then the functional μX : Y → χ(X ∩ Y ) is a valuation provided Y is restricted appropriately. This valuation is not smooth, but a general smooth valuation can be approximated by linear combinations of valuations of this type. Furthermore it is natural to define the restricted product μX · μY := μX∩Y whenever the intersection is nice enough, and the Alesker product of smooth valuations is the natural extension. These remarks underlie the following. Proposition 2.4.3. Let X ⊂ M be a smooth submanifold with corners, and let {Ft }t∈P be a smooth proper family of diffeomorphisms of M . Thus the induced maps on the cosphere bundle yield a smooth map F˜ : P ×S ∗ M → S ∗ M . Assume for each ξ ∈ S ∗ M the property that the induced map F˜ξ : P → S ∗ M is a submersion, and let dm be a smooth measure on the parameter space P . Then     χ Ft (X) ∩ Y dm (2.4.2) X, {Ft }t∈P , dm (Y ) := P

determines an element of V(M ), and the span of all such valuations is dense in V(M ) in an appropriate sense. Furthermore        χ Ft (X)∩Gs (Y )∩Z dn(s) dm(t). X, {Ft }t , dm · Y, {Gs }s∈Q , dn (Z) = P

Q

(2.4.3) While the right-hand side of (2.4.3) is well defined under the given conditions —this follows from an argument similar to the corresponding part of the proof of

2.4. Valuations and integral geometry on isotropic manifolds

81

Theorem 2.2.4— the resulting smooth valuation does not have the same form. An important instance of the construction (2.4.2) arises if P = G is a Lie group acting isotropically on M and dm = dg is a Haar measure. Alesker has observed that the discussion of Section 2.1.5, together with the Nash embedding theorem (or, more simply, local smooth isometric embedding of Riemannian manifolds), shows that given any smooth n-dimensional Riemannian manifold M there is a canonical Lipschitz–Killing subalgebra LK(M )  R[t]/(tn+1 ) of V(M ), obtained by isometrically embedding M in a Euclidean space and restricting the resulting intrinsic volumes to M .

2.4.2 First variation, the Rumin operator, and the kernel theorem Let M n be a connected smooth oriented manifold and μ ∈ V(M ). Given a vector field V on M , denote by Ft : M → M the flow generated by V . We consider the first variation of μ with respect to V , given by d μ(Ft (A)), (2.4.4) δV μ(A) := dt t=0 where A ⊂ M is nice. Clearly μ = 0 if and only if δμ = 0 and μ({p}) = 0 for some point p ∈ M . If μ = Ψ(θ,ϕ) ∈ Ωn (M ) × Ωn−1 (S ∗ M ), then the first variation may be rep∗ resented as follows. Put F˜t := F−t : S ∗ M → S ∗ M for the corresponding flow of contact transformations of the cosphere bundle. Then F˜ is the flow of a vector field V˜ on S ∗ M . One has that ) (  d θ+ ϕ δV μ(A) = dt t=0 Ft (A) N ∗ (Ft (A)) ) (  d = θ+ ϕ dt t=0 Ft (A) F˜t N ∗ (A) ) (  d F˜t∗ ϕ = F ∗θ + dt t=0 A t N ∗ (A)   d d ∗ F θ + (2.4.5) = F˜t∗ ϕ t dt dt ∗ A N (A) t=0 t=0   = LV θ + LV˜ ϕ A N ∗ (A)   diV θ + (diV˜ + iV˜ d)ϕ = A

N ∗ (A)

82

Chapter 2. Algebraic Integral Geometry  = N ∗ (A)

iV˜ (π ∗ θ + dϕ),

since π∗ N (A) = ∂[[A]] and ∂N (A) = 0. In particular, this last expression is independent of the choice of differential forms θ and ϕ representing μ. Following Bernig and Br¨ ocker [23], this criterion becomes much clearer and more useful with the introduction of the Rumin differential D. Recall that if α is a contact form on a contact manifold M , then there exists a unique Reeb vector field T such that iT α ≡ 1, LT α ≡ 0, iT dα ≡ 0. (Of course these three conditions are redundant.) If M = SRn , then T(x,v) = (v, 0). Let Q := α⊥ denote the contact distribution, which carries a natural (up to scale) symplectic structure given by dα. Recall that a differential form on a contact manifold is said to be vertical if it is a multiple of the contact form. Proposition 2.4.4 ([55]). Let S 2n−1 be a contact manifold and ϕ ∈ Ωn−1 (S). Then there exists a unique vertical form α ∧ ψ such that d(ϕ + α ∧ ψ) is vertical. Proof. Observe that given the choice of α there is a natural injection j from sections of Λ∗ Q∗ to Ω∗ S, determined by the conditions (i) that j followed by the restriction to Q is the identity and (ii) that iT ◦ j = 0. In fact the image of j is precisely the subspace of forms annihilated by iT , as well as the image of iT . We compute, for general ψ,         d ϕ + α ∧ ψ = α ∧ iT d ϕ + α ∧ ψ + iT α ∧ d ϕ + α ∧ ψ    ≡ iT α ∧ dϕ + dα ∧ ψ mod α. Thus ψ will satisfy our requirements if and only if   dϕ + dα ∧ ψ Q ≡ 0.

(2.4.6)

But it is not hard to prove the following (actually a consequence of the fact that multiplication by ω is a Lefschetz operator in an sl(2) structure on Λ∗ Q): Fact 2.4.5. Suppose (Q2n−2 , ω) is a symplectic vector space. Then (1) Multiplication by ω yields a linear isomorphism Λn−2 Q → Λn Q. (2) Multiplication by ω 2 yields a linear isomorphism Λn−3 Q → Λn+1 Q. The restriction of dα to Q is a symplectic form. Applying this pointwise and using the observations above we find that there is a form ψ, uniquely defined modulo α, such that (2.4.6) holds.  We define the Rumin differential of ϕ to be   Dϕ := d ϕ + α ∧ ψ . It is clear that D annihilates all multiples of α and of dα.

2.4. Valuations and integral geometry on isotropic manifolds

83

Lemma 2.4.6. ω ∧ iT Dϕ = 0. Proof. This is equivalent to the relation dα ∧ Dϕ = 0. But   dα ∧ Dϕ = d α ∧ Dϕ = 0, since α ∧ Dϕ = 0 by construction.



Now we can rewrite (2.4.5) as    δV μ(A) = iV˜ π ∗ θ + Dϕ N ∗ (A)      π ∗ iV θ + iV˜ α ∧ iT Dϕ = N ∗ (A)    π ∗ iV θ + iV˜ (α) ∧ iT Dϕ = N ∗ (A)    = π ∗ iV θ + α, V  ∧ iT Dϕ,

(2.4.7)

N ∗ (A)

since N ∗ (A) is Legendrian. Since this expression does not involve derivatives of V , this shows that the first variation operator takes values among covector-valued curvature measures. Theorem 2.4.7 (Bernig–Br¨ ocker [23]). The valuation Ψθ,ϕ is equal to 0 if and only  if π ∗ θ + Dϕ = 0 and S ∗ M ϕ = 0 for some point p ∈ M . p

Proof. To prove “if”, by finite additivity it is enough to prove that Ψθ,ϕ (A) = 0 for contractible sets A. In this case, if Ft is the flow of a smooth vector field V contracting A to a point p, then N (Ft (A)) → N ({p}) = [[Sp∗ M ]]. Hence, by (2.4.7),  Ψθ,ϕ (A) = Ψθ,ϕ ({p}) =

ϕ = 0. Sp∗ M

By Proposition 2.2.3, to prove the converse it is enough to show that if (π ∗ (iV θ) + α, V  ∧ iT Dϕ)|Q ≡ 0 mod dα for all V , then π ∗ θ + Dϕ = 0. But this follows from conclusion (2) of Fact 2.4.5 and Lemma 2.4.6.  Since the normal cycle is closed and Legendrian, it is immediate that any multiple of the contact form (i.e., a “vertical” form with respect to the contact structure of S ∗ W ), or any exact form, yields the zero valuation. Corollary 2.4.8. If M is contractible, then Ψ0,ϕ = 0 if and only if ϕ lies in the linear span of the vertical forms and the exact forms. This characterization also implies conclusion (2.4.2) of Theorem 2.4.2: if ψ = Ψθ,ϕ ∈ V G (M ), then the kernel theorem implies that δψ ∼ π ∗ θ+Dϕ ∈ Ωn (S ∗ M )G .

84

Chapter 2. Algebraic Integral Geometry

But this space is finite-dimensional, and (by the kernel theorem again) the kernel of the first variation map δ is the one-dimensional space χ. As another application, observe that in the presence of a Riemannian struc with a scalarture on M we may identify a covector-valued curvature measure ψ valued one ψ by putting 

 f dψ A :=



A f nA , dψ



for smooth domains A, where nA is the outward pointing normal. The first variation may thus be expressed as a scalar-valued curvature measure δμ given by 

 A f d δμ = δf nA μ(A).

Now if M is a Euclidean space then we say that a curvature measure ψ ≥ 0 if the measure ψ A ≥ 0 whenever A ∈ K. Theorem 2.4.9. A valuation μ ∈ Valsm (Rn ) is monotone if and only if δμ ≥ 0 and μ({pt}) ≥ 0.  Since the first variation operator is a graded map of degree −1 from Val(Rn ) to translation-invariant curvature measures, we obtain Corollary 2.4.10. μ ∈ Val(Rn ) is monotone if and only if all of its homogeneous components are monotone. Proof. This is easily proved by showing that a translation-invariant curvature measure is nonnegative if and only if its homogeneous components are all nonnegative.  Finally, the Rumin operator also figures prominently in a remarkable formula of Alesker–Bernig for the product of two smooth valuations in terms of differential forms representing them. We will only need the following general characterization. Theorem 2.4.11 (Alesker–Bernig [12]). For i = 1, 2, the product of two smooth valuations Ψθi ,ϕi may be expressed as Ψθ0 ,ϕ0 , where θ0 and ϕ0 may be computed from the θi and ϕi using only the Rumin differential D and canonical fiber integrals.  It is interesting to note that the expression for Ψθ0 ,ϕ0 given in [12] is not symmetric in i = 1, 2. In fact it gives a proof of the following observation of Bernig: Corollary 2.4.12. The space of smooth curvature measures is a module over the algebra of smooth valuations.

2.4. Valuations and integral geometry on isotropic manifolds

85

2.4.3 The filtration and the transfer principle for valuations Alesker [9] showed that the algebra V(M n ) admits a natural filtration V(M ) = V0 (M ) ⊃ · · · ⊃ Vn (M ) that respects the product. The filtration index of a valuation μ may be expressed as the largest value of i such that μ may be represented by differential forms θ and ϕ, where in some family of local trivializations of S ∗ M all terms of the differential form ϕ involve at least i variables from the coordinates of the base space. Thus Vn (M ) = {Ψθ,0 : θ ∈ Ωn (M )} is the space n of smooth signed measures on M . The associated graded algebra gr(V(M )) = i=0 Vi (M )/Vi+1 (M ) is naturally isomorphic to the algebra Γ(Valsm (T M )) of sections of the infinitedimensional vector bundle Valsm (T M ) → M . A valuation ϕ ∈ Vk (M ) may be thought of as defining a “k-dimensional volume” of M : for N k ⊂ M ,    Kl[ϕ]x Tx N d volk x, (2.4.8) ϕ(N k ) = N

where [ϕ] ∈ Γ(Val (T M )) corresponds to the equivalence class of the valuation ϕ in Vk (M )/Vk+1 (M ). Reducing further, this value in fact only depends on the even parts of the [ϕ]x ∈ Valsm k (Tx M ). In this formula, ϕ may be thought of as operating on any compact C 1 submanifold of M , or more generally on any appropriately rectifiable set of the same dimension. If G acts isotropically on M , then this gives rise to a transfer principle at the level of first-order formulas, similar to (2.1.34), as follows. The isomorphism above gives rise to the restriction sm

 G     ValH To M , grV G (M )  Γ Valsm T M where H ⊂ G is the subgroup stabilizing the representative point o. Thus, there is a canonical map V G (M ) → ValH (To M ); for ϕ ∈ V G (M ) we put [ϕ] ∈ ValH (To M ) for its image under this map. If ψ ∈ Vk+1 , then ψ(V ) = 0 whenever V ⊂ M is a smooth submanifold of dimension k. Thus if [ϕ] ∈ ValH k (To M ), then [ϕ](V ) = ϕ(V ) is well defined, and in fact may be expressed as    Kl[ϕ] Tx V dx, (2.4.9) [ϕ](V ) = V

where Tx V ⊂ Tx M is identified via the group action with a subspace of To M . Obviously, H acts transitively on the sphere of To M . Thus ValH ⊂ Valsm + by Proposition 2.3.17, and we may consider the kinematic operator       (2.4.10) kH : ValH To M −→ ValH To M ⊗ ValH To M . Furthermore, any element φ ∈ ValH k (To M ) acts on k-dimensional submanifolds V k by

86

Chapter 2. Algebraic Integral Geometry

Theorem 2.4.13. In the setting above, let V k , W l ⊂ M be compact C 1 submanG (M ) correspond to [ϕ] ∈ ValH ifolds, and let ϕ ∈ Vk+l−n k+l−n,+ (To M ) as above. Then       (2.4.11) ϕ V ∩ gW dg = kH ϕ V, W . G

Here the elements of ValH (To M ) are evaluated on V and W as in the discussion above, noting that since kH is a graded operator it is only necessary to evaluate valuations of degree k and l on V and W , respectively. In other words, at the level of first-order formulas the kinematic formulas for ¯ are identical. Since the space of invariant valuations on a (M, G) and (To M, H) vector space is graded, and the Klain map on each graded component is injective, the first-order formulas there carry the same information as the full kinematic operator kH . However, the spaces V G (M ) are only filtered, so one loses information in passing from the full kinematic operator kG to the first-order version.

2.4.4 The FTAIG for compact isotropic spaces 

Lemma 2.4.14.

  ϕ gA ∩ · dg.

μG A ·ϕ=

(2.4.12)

G

Proof. Restricting to valuations of the form considered there, this follows from Proposition 2.4.3.  Theorem 2.4.15 ([12, 24, 40]). Let (M, G) be a compact isotropic Riemannian manifold. Put V G∗ (M ) for the dual space to the finite-dimensional vector space V G (M ) and p : V G (M ) → V G∗ (M ) for the Poincar´e duality map of Theorem 2.4.2; put m∗G : V G∗ (M ) → V G∗ (M ) ⊗ V G∗ (M ) for the adjoint of the restricted Alesker product map. Then the following diagram commutes: V G (M )

vol(M )−1 kG

p

 V G∗ (M )

/ V G (M ) ⊗ V G (M )

(2.4.13)

p⊗p

m∗ G

 / V G∗ (M ) ⊗ V G∗ (M ).

G G Proof. Given a smooth polyhedron A ⊂ M , define μG A ∈ V (M ) by μA :=  G μ(gA) dg, i.e., the invariant valuation μA (B) := G χ(gA ∩ B) dg, where dg G is the Haar measure with the usual normalization. As in the translation-invariant case, these valuations span V G (M ). Using Lemma 2.4.3, to prove (2.4.13) it is enough to show that     G   ∗   G  G p ⊗ p ◦ kG μG μB ⊗ μG μB ⊗ μG (2.4.14) A C = vol(M ) mG ◦ p μA C

for all polyhedra A, B, and C.

2.4. Valuations and integral geometry on isotropic manifolds

87

Direct calculation from (2.4.12) implies that p(μG A ) = vol(M ) evA , where evA denotes the evaluation at A functional on V G (M ). Using the fact that p is self-adjoint it is now easy to verify that both sides of (2.4.14) are equal to G G  vol(M )2 (μG A · μB · μC )(M ).

2.4.5 Analytic continuation As in Section 2.2.4, some isotropic spaces come in families, indexed both by dimension and by curvature. The main examples are the real space forms (spheres, Euclidean spaces, and hyperbolic spaces), the complex space forms (CP n and Cn under the action of the holomorphic Euclidean group U (n), and CH n ) and the quaternionic space forms (HP n and Hn under the action of the quaternionic Euclidean group Sp(n) × Sp(1), and HH n ). There are also octonion versions of these families in octonion dimensions 1 and 2, the 2-dimensional projective case being the Cayley plane. We already have one method —the transfer principle— for comparing the integral geometry within each of these families. Another method arises from a more geometric perspective, viewing the curvature λ as a parameter in the various kinematic and product formulas. Fortunately, all of the formulas are analytic in λ, so we may extend results from the compact λ > 0 cases to the noncompact cases λ ≤ 0, where direct calculations are more difficult. The model case is that of the real space forms Mλ . The transfer principle shows that the kinematic formulas are the same at the level of curvature measures if the latter are identified via their correspondence with invariant differential forms. In the cases λ = 0, 1 we have seen in Theorems 2.3.16 and 2.4.15 that the kinematic formulas are related to the Alesker product via Poincar´e duality, although a priori this duality has different meanings in the two cases. Clearly the same is true for all λ ≥ 0. What about the cases λ < 0? On the face of it there can be no FTAIG in the hyperbolic case: the invariant valuations are obviously not compactly supported, and the natural Poincar´e duality [9] for smooth valuations on a manifold pairs a general valuation with a valuation with compact support. Nevertheless, such a pairing does exist for invariant valuations: let τi ∈ V Gλ (Mλ ) denote the invariant valuation corresponding to ti ∈ V G0 (Rn ) = ValSO(n) (Rn ) under the transfer principle, i = 0, . . . , n (this is possible since in this case the correspondence between invariant curvature measures and invariant valuations is bijective). Let {τi∗ } denote the dual basis. Rescaling as necessary in the cases λ > 0, the FTAIG may be restated in a unified way for all λ ≥ 0 by taking the Poincar´e duality map to be     p(φ), ψ := τn∗ φ · ψ for φ, ψ ∈ V Gλ (Mλ ) and λ ≥ 0. Now analytic continuation shows that the same form of the FTAIG holds also for λ < 0. To carry this out, we construct a common model for the space

88

Chapter 2. Algebraic Integral Geometry

of invariant valuations on all of the Mλ , wherein the Alesker products, kinematic operators and Poincar´e pairings vary polynomially with λ. The Lie algebras gλ of (2.2.12) may be represented by a family of Lie brackets on the common vector space Rn ⊕ so(n) by putting     (v, h), (w, j) λ := hw − jv, [h, j] + λ v ⊗ w − w ⊗ v . The tangent spaces To¯SMλ are represented by Rn × so(n)/so(n − 1)  Rn ⊕ Rn−1 , and in these terms the transfer map of Theorem 2.2.5 is the identity. In particular, the pullback of the contact form to Tid Gλ = gλ is α(v, h) = v1 , independent of λ. Therefore, the Maurer–Cartan equations imply that the exterior derivatives dλ on Ω∗Gλ (SMλ ) are represented by a polynomial family of operators Λ∗ (Rn ⊕ Rn−1 )SO(n−1) −→ Λ∗+1 (Rn ⊕ Rn−1 )SO(n−1) . Furthermore, the uniqueness statement in Proposition 2.4.4 implies that the same is true of the Rumin derivatives Dλ . Theorem 2.4.11 now implies that the Alesker product also varies polynomially, which completes the proof. The discussion above is simplified somewhat by the fact that the map from invariant curvature measures to invariant valuations is injective for the real space forms. Nevertheless, it extends also to the 1-parameter families of complex, quaternionic, and octonionic space forms via the following general statement. Theorem 2.4.16. Let Gλ be a 1-parameter family of Lie groups acting isotropically on spaces Mλ , with the following properties: (1) The associated Lie algebras gλ are given by a family [ · , · ]λ of Lie brackets, depending analytically on the parameter λ, on a fixed finite-dimensional real vector space g. (2) The Gλ have a common subgroup H such that Mλ = Gλ /H, and the restriction of [ · , · ]λ to the associated Lie algebra h ⊂ g is constant. (3) The adjoint action of h on gλ is independent of λ. (4) If λ > 0, then Mλ is compact. Then there exists a family of elements fλ ∈ V Gλ ∗ (Mλ ) determined by the condition   fλ Ψϕ,c d volMλ ≡ c (2.4.15) whenever ϕ ∈ Ωn−1,Gλ (SMλ ), such that, putting     p = pλ : V Gλ (Mλ ) −→ V Gλ ∗ (Mλ ) by p(φ), ψ := fλ φ · ψ ,

(2.4.16)

M = Mλ and G = Gλ , diagram (2.4.13) commutes for all λ. Proof. Since To Mλ  g/h for the distinguished point o = oλ ∈ Mλ (i.e., the image of the identity element of Gλ ), condition (3) implies that the identity map of g induces isometries (2.2.6) between these spaces, intertwining the action of H, for the

2.4. Valuations and integral geometry on isotropic manifolds

89

various values of λ. The proof of Theorem 2.2.5 implies that the canonical 1-forms of the SMλ and their differentials correspond to fixed elements α ∈ Λ1 (g/k)K and ω ∈ Λ2 (g/k)K , respectively, independent of λ. Furthermore, Theorem 2.2.5 implies  that the identification Λ0 := Λn (g/h)H ⊕ Λn−1 (g/k)K /(α, ω)  CurvGλ (Mλ ) is independent of λ, as is the coproduct structure on Λ0 induced by the k˜λ = k˜Gλ . Here, as usual, K ⊂ H is the isotropy subgroup of the distinguished point o¯ = o¯λ ∈ SMλ and k ⊂ h is its Lie algebra. By the reasoning above, the Rumin operators of SMλ induce an analytic family of operators Dλ : Λn−1 (g/k)K /(α, ω) → Λn (g/k)K , and by the kernel theorem (Theorem 2.4.7), the kernel of the map Ψλ : Λ0 → V Gλ (Mλ ) is the same as that of the map Δλ : Λ0 −→ Λ1 := Λn−1 (h/k)K ⊕ Λn (g/k)K given by Δλ (β, γ) := (r(γ), π ∗ β + Dλ γ), where r : Λ∗ (g/k) → Λ∗ (h/k) is the restriction map. Now choose a subspace W ⊂ Λ0 such that the restriction of Ψλ gives a linear ˜ λ } of isomorphism Ψ λ : W → V Gλ (Mλ ) locally in λ, and consider the family {m compositions W ⊗W

Ψλ ⊗Ψλ

/ V Gλ (Mλ ) ⊗ V Gλ (Mλ )

mλ

/ V Gλ (Mλ )

Ψ−1 λ

/ W.

We claim that this family is analytic in λ. To see this we observe that m ˜ λ can also be expressed as the composition W ⊗W

i

/ Λ0 ⊗ Λ0

m ¯λ

/ Λ0 = kerλ ⊕W

πλ

/ W,

where i is the inclusion, m ¯ λ is the map induced by the construction of Theorem 2.4.11, kerλ is the kernel of Ψλ , and πλ is the projection with respect to the given decomposition. As in the discussion above, m ¯ λ is analytic in λ, so it remains to show that the same is true for πλ . But kerλ is also the kernel of Δλ : Λ0 → Λ1 . Thus we may write Λ1 = U ⊕ V , where the composition of Δλ |W with the projection to U is an isomorphism locally in λ, which is clearly analytic in λ. Hence the same is true for the inverse of this composition, and πλ is the composition of the projection to U with this inverse, which is thereby again analytic. The proof is concluded by applying analytic continuation, noting that by condition (4) the Poincar´e duality maps pλ have the given form (2.4.15) and (2.4.16) for λ > 0.  Another sticky technical point concerns the extension of the Crofton formulas (2.1.27) to the hyperbolic case, where the spaces of totally geodesic submanifolds of various dimensions replace the affine Grassmannians. In the spherical

90

Chapter 2. Algebraic Integral Geometry

cases the corresponding formulas are literally special cases of the kinematic formula, where the moving body is taken to be a totally geodesic sphere. In the Euclidean case these formulas may be regarded as limiting cases of kinematic formulas where the moving body is taken to be a ball of the given dimension and of radius R → ∞: for finite R there are boundary terms, but these become vanishingly small in comparison with the volume in the limit. However, neither of these devices are available in the hyperbolic cases. Nevertheless, the formulas may be analytically continued. Theorem 2.4.17. Let Grλ (n, k) denote the Grassmannian of all totally geodesic submanifolds of dimension k in Mλn . If we put      P¯ , φλ (A) := χ A ∩ P¯ dmn−1 λ Grλ (n,n−1) n

where dmkλ is the Haar measure on Grλ (n, k), then φλ ∈ V Gλ (Mλn ). Furthermore, if these measures are normalized appropriately, then  n      ¯ dmkλ (Q) ¯ = τk + χ · ∩Q pki λ τi , (2.4.17) φkλ = Grλ (n,n−k)

i>k

where each pki is a polynomial. Proof. Consider the pullback to Rn of the standard Riemannian metric on the n of radius R under the spherical projection map sphere SR (p, R) p −→ R * . |p|2 + R2

n Rn −→ SR ,

In terms of polar coordinates R+ × S n−1 → Rn , (r, v) → rv, one computes the pullback metric to be 

dr2 1+

r2 R2

2 +

r2 dv 2 dr2 r2 dv 2 = + , 2 2 r2 (1 + λr ) 1 + λr2 1 + R2

where dv 2 is the usual metric on the unit sphere. (Of course these coordinates cover only the northern hemisphere.) Meanwhile, the Hilbert metric on B(0, R), which yields the Klein model for H n with curvature −1, may be expressed as ds2Hilbert =



dr2

R2 1 −

 + r2 2

R2

r2 dv 2  2 . R2 1 − Rr 2

(2.4.18)

Scaling this up by a factor of R2 yields 

dr2 1−

r2 R2

2 + 

dr2 r2 dv 2 r2 dv 2 = + =: ds2λ , 2 2 r2 (1 + λr ) 1 + λr2 1 − R2

(2.4.19)

2.4. Valuations and integral geometry on isotropic manifolds

91

where again λ = −R−2 is the curvature. In these coordinates all of the various totally geodesic hypersurfaces appear as hyperplanes (or their intersections with the disk) of Rn . In other words, the codimension 1 affine Grassmannian Gr := Grn−1 (Rn ) of Rn is a model for all of the various affine Grassmannians as the curvature λ varies. We again parametrize Gr by polar coordinates, i.e., put P¯ (r, v) := {x : v · x = r}. Then dmλ =

dr dv (1 + λr2 )

n+1 2

.

(2.4.20)

This is easily checked for λ ≥ 0, and is a direct calculation if λ < 0 and n = 2. If λ < 0, note that this measure assigns the value 0 to any set of hyperplanes that does not meet the relevant ball.  The complex space forms also fit together nicely as a one-parameter family. Adapting well-known formulas for the Fubini–Study metric ([50, p. 169]) we may define the Hermitian metrics   

 

1 + λ|z|2 ds2C − λ z¯i dzi zi d¯ zi 2 dsC,λ :=  2 1 + λ|z|2 √ for all λ ∈ R, defined on all of Cn if λ ≥ 0 and on the ball B(0, 1/ −λ) ⊂ Cn if λ < 0, where ds2C denotes the standard Hermitian metric on Cn . This yields a metric of constant holomorphic sectional curvature 4λ on an affine coordinate patch (covering the whole space if λ ≤ 0) of the corresponding complex space form. Exercise 2.4.18. Complete this discussion by considering the invariant measures on the spaces of totally geodesic complex submanifolds, and also on totally geodesic isotropic submanifolds, i.e., submanifolds on which the K¨ahler form vanishes. These are the unitary orbits of Rk , RP k and RH k (0 ≤ k ≤ n) in Cn , CP n and CH n , respectively. Exercise 2.4.19. Do the same for the quaternionic space forms.

2.4.6 Integral geometry of real space forms We use the transfer principle and analytic continuation to study the isotropic spaces  n      R , SO(n) , S n , SO(n + 1) , and H n , SO(n, 1) . For simply connected spaces Mλn of constant curvature λ, put τi for the valuations corresponding to ti under the transfer principle. These are not the Lipschitz–Killing valuations, but they do correspond to appropriate multiples of the

92

Chapter 2. Algebraic Integral Geometry

elementary symmetric functions of the principal curvatures for hypersurfaces. Nijenhuis’s theorem and the transfer principle yield  αn kMλn (τl ) = n+1 τi ⊗ τj , l = 0, . . . , n. (2.4.21) 2 i+j=n+l

Taking λ > 0, the values of the τi on the totally geodesic spheres S j (λ) ⊂ S n (λ) = Mλn are i    2 . τi S j (λ) = δji · 2 √ λ Therefore,

 2   i  λ n

χ=

i=0

4

τ2i

(2.4.22)

in M n (λ) for λ > 0, and by analytic continuation this formula holds for λ ≤ 0 as well. Therefore, the principal kinematic formula in Mλn may be written as  2   i  λ n

kMλn (χ) =

4

i=0

αn  kMλn (τ2i ) = n+1 τi ⊗ 2 i



∞  j  λ j=0

4

 τn−i+2j .

(2.4.23)

Returning to the case λ > 0, let φ denote the invariant valuation given √ by 2/ λ times the average Euler characteristic of the intersection with a totally geodesic hypersphere. Then, using the standard normalization (2.1.29) for the Haar measure on SO(n + 1),  k

φ := k

=

2 √ λ

2 λ

k

1

λ2n αn

1 2 (n−k)



  χ · ∩gS n−k (λ) dg SO(n+1)

  kS n (λ) (χ) · , S n−k (λ)

αn ∞  j  λ = τ2j+k , 4 j=0

(2.4.24)

k = 0, . . . , n.

In particular, if N k is a compact piece of a totally geodesic submanifold of dimension k, then 2k+1 φk (N ) = τk (N ) = |N |. αk Comparing (2.4.22) and (2.4.24) we find that χ = τ0 +

λ 2 φ 4

(2.4.25)

2.4. Valuations and integral geometry on isotropic manifolds or

λ 4

τ0 = χ −

93

 χ( · ∩ H) dH.

(2.4.26)

Grλ,n−2

This formula is due to Teufel [61] and Solanes [58]. Now the principal kinematic formula (2.4.23) may be written as αn  τi ⊗ φ j . kMλn (χ) = n+1 2 i+j=n

(2.4.27)

By the multiplicative property, kMλn (τk ) =

  αn  τi ⊗ φ j · τk . 2n+1 i+j=n

(2.4.28)

Comparing with (2.4.21) we find that φj · τi = τi+j .

(2.4.29)

This is the reproductive property of the τi . At this point we may write    kMλn (ψ) = (ψ ⊗ τ0 ) · φi ⊗ φj

(2.4.30)

i+j=n

whenever ψ is an invariant valuation on Mλn . By analytic continuation these formulas are valid for λ ≤ 0 as well, where φ is the corresponding integral over the space of totally geodesic hyperplanes, normalized so that φ(γ) = π2 |γ| for curves γ. By (2.4.24), the valuations φ, τ1 and the generator t of the Lipschitz–Killing algebra all coincide if the curvature λ = 0. For general λ, the relations (2.4.25) and (2.4.29) yield λ τi = φi − φi+2 . (2.4.31) 4 There is also a simple general relation between φ and t. To put it in context, denote by Vλn the algebra of invariant valuations on Mλn . Since each Mλn embeds essentially uniquely into Mλn+1 , and every isometry of Mλn extends to an isometry on Mλn+1 , there is a natural surjective restriction homomorphism Vλn+1 → Vλn . Put Vλ∞ for the inverse limit of this system. Thus Vλ∞ is isomorphic to the field of formal power series in one variable, which may be taken to be either φ or t. The valuations τi may also be regarded as living in Vλ∞ , since they behave well under the restriction maps. The relations among valuations given above are also valid in Vλ∞ , except for those that depend explicitly on the kinematic operators. These depend on the dimension n and definitely do not lift to Vλ∞ . Proposition 2.4.20. t= +

φ 1−

λφ2 4

,

φ= +

t 1+

λt2 4

.

(2.4.32)

94

Chapter 2. Algebraic Integral Geometry

Proof. We use the template method to prove that φ2

2

t =

1−

λφ2 4

 =φ

2

λ 1 + φ2 + 4

 2  λ 4 φ + ··· 4

(2.4.33)

for λ > 0, so t must be given by the square root that assigns positive values (lengths) to curves. The generalization to all λ ∈ R follows by analytic continuation. Since everything scales correctly, it is enough to check this for λ = 1. In fact, it is enough to check that the values of the right- and left-hand sides of (2.4.33) agree on spheres S 2l of even dimension. Clearly  2 · 4k , if k ≤ l, 2k 2l φ (S ) = 0, if k > l, so the right-hand side yields 8l. To evaluate the left-hand at S 2l we recall that   4ω2    4  t2 (S 2l ) = 2t2 B 2l+1 = 2 μ2 B 2l+1 = μ2 B 2l+1 . π π Meanwhile, Theorem 2.1.1 for the volume of the r-tube about B 2l+1 yields    ω2l−i+1 μi B 2l+1 r2l+1−i . ω2l+1 (1 + r)2l+1 = Equating the coefficients of r2l−1 we obtain   ω2l+1 2l + 1 2l+1 = 2πl, )= μ2 (B 2 ω2l−1 in view of the identity 2π ωn . = ωn−2 n

2.5



Hermitian integral geometry

Our next goal is to work out the kinematic formulas for Cn under the action of the unitary group U (n). The starting point is Alesker’s calculation of the Betti numbers and generators of ValU (n) (Cn ), which we state in a slightly stronger form. Let DCn denote the unit disk in Cn . U (n)

Theorem 2.5.1 ([4]). ValU (n) (Cn ) is generated by t ∈ Val1  U (n) χ( · ∩ P¯ ) dP¯ ∈ Val2 , s := C

Grn−1

and (2.5.1)

2.5. Hermitian integral geometry

95

C

where Grn−1 is the codimension 1 complex affine Grassmannian and the Haar measure dP¯ is normalized so that s(DC1 ) = 1. There are no relations between s and t in (weighted) degrees ≤ n. U (n)

U (n)

By Alesker–Poincar´e duality, dim Val2n−k = dim Valk , so this determines the Betti numbers completely. In fact, the Poincar´e series for ValU (n) is 

U (n)

dim Vali

xi =

(1 − xn+1 )(1 − xn+2 ) =: σ(x). (1 − x)(1 − x2 )

(2.5.2)

This follows at once from the palindromic nature of σ(x), which may be expressed as the identity x2n σ(x−1 ) = σ(x). By (2.3.4) and Theorem 2.3.4, sk tl (A) = c

 ¯C Gr n−k

μl (A ∩ P¯ ) dP¯

for some constant c. Exercise 2.5.2. Show that  k tl ) = csn−k ∗ t2n−l (s and  k tl )(A) = c (s

 GrC k

  μ2n−2k−l πE (A) dE,

(2.5.3) A ∈ K(Cn )

(2.5.4)

for some constants c. In fact, Alesker [4] showed that {sk tl : k ≤ min(k + 2l , n − k − 2l )} is a basis for ValU (n) , as is its Fourier transform.

2.5.1 Algebra structure of ValU (n) (Cn ) The next step in unwinding the integral geometry of the complex space forms is Theorem 2.5.3 ([40]). Let s and t be variables with formal degrees 2 and 1, respectively. Then the graded algebra ValU (n) (Cn ) is isomorphic to the quotient R[s, t]/(fn+1 , fn+2 ), where fk is the sum of the terms of weighted degree k in the formal power series log(1 + s + t). The fk satisfy the relations f1 = t, t2 , 2 ksfk + (k + 1)tfk+1 + (k + 2)fk+2 = 0, f2 = s −

k ≥ 1.

(2.5.5)

96

Chapter 2. Algebraic Integral Geometry

Proof. For general algebraic reasons (viz. (2.5.2) and Alesker–Poincar´e duality), the ideal of relations is generated by independent homogeneous generators in degrees n + 1 and n + 2. Since the natural restriction map ValU (n+1) (Cn+1 ) −→ ValU (n) (Cn ) is an algebra homomorphism, and maps s to s and t to t, it is enough to show that fn+1 (s, t) = 0 in ValU (n) . To prove this, we use the template method and the transfer principle. Let n n GrC n−1 (CP ) denote the Grassmannian of C-hyperplanes P ⊂ CP , equipped with the Haar probability measure dP , and define  U (n+1) χ( · ∩ P ) dP ∈ V2 (CP n ). (2.5.6) s¯ := n GrC n−1 (CP )

It is not hard to see that the valuations s, t ∈ ValU (n) (Cn ) correspond via the transfer principle (Theorem 2.4.13) to s¯, t ∈ V U (n+1) (CP n ) modulo filtration 3 and 2, respectively. Since   2n πn =⇒ t2n (CP n ) = t2n (DnC ) = vol(CP n ) = vol(DCn ) = n n! by (2.3.27), it follows from Section 2.4.3 that sk t2n−2k (DCn ) = s¯k t2n−2k (CP n ) = t2n−2k (CP n−k ) =

  2n − 2k , n−k

k = 0, . . . , n.

(2.5.7) The following lemma and its proof were communicated to me by I. Gessel [42]. I am informed that it is a special case of the “Pfaff–Saalsch¨ utz identities.” Lemma 2.5.4.  n+1 2 

 i=0

     n + 1 − i 2n − 2k − 2i k (−1)i (−1)n−k = . n−k−i i n−k n+1−i n+1

(2.5.8)

In particular, the left-hand side is zero if 2k < n. Proof. In terms of generating functions, the identity (2.5.8) that we desire may be written as  (−1)i (n + 1) n + 1 − i2n − 2k − 2i 1 . (2.5.9) xn y k = n−k−i i n+1−i 1 − xy(1 − x) n,k,i

But using well-known generating functions [64], the left-hand side may be expressed as the sum of        2n − 2k − 2i n k  m + k m+k+i k i n−i i 2m (−1) (−1) y x y = x n−k−i i−1 i−1 m n,k,i

m,k,i

2.5. Hermitian integral geometry

97 =−

2m

(1−x)m+k xm+k+1 y k m m,k 2m  = −x (1−x)k xk y y (1−x)m xm m m k

−x

=* 1 − 4x(1 − x)(1 − xy(1 − x)) and 

 (−1)

i

n,k,i

n+1−i i



     2n−2k−2i n k m+k m+k+i k i 2m (−1) y x y = x n−k−i i−1 m m,k,i     m+k+1 m+k+i k i 2m = (−1) y x i m m,k,i

1−x , =* 1 − 4x(1 − x)(1 − xy(1 − x)) hence the sum is *

1 − 2x 1 − 4x(1 − x)(1 − xy(1 − x))

=

1 , 1 − xy(1 − x) 

as claimed. U (n)

n

With (2.5.7), the lemma yields the following identities in Val (C ): ⎞ ⎛ n+1  2   (−1)i n + 1 − i tn−2k−1 sk · ⎝ tn−2i+1 si ⎠ = 0, 0 ≤ 2k < n. (2.5.10) i n + 1 − i i=0 U (n)

Since the tn−2k−1 sk in this range constitute a basis of Valn−1 (Cn ), Alesker– Poincar´e duality implies that the sum in the second factor is zero. This sum is  (−1)n fn+1 (s, t). Strictly speaking, we have now done enough to determine the kinematic operator kU (n) in terms of the first Alesker basis {sk tl }: by the remarks following Theorem 2.3.16, the coefficients are the entries of the inverses of Poincar´e pairing matrices determined   by (2.5.7). The latter are Hankel matrices with ascending enıve expansion tries of the form 2k k . However, the expressions resulting from the na¨ for the inverse seem unreasonably complicated (although perusal of a few of them hints at an elusive structure). A more useful formulation will be given below.

2.5.2 Hermitian intrinsic volumes and special cones Every element of ValU (n) (Cn ) is a constant coefficient valuation, as follows. Let (z1 , . . . , zn , ζ1 , . . . , ζn ) be canonical coordinates on T Cn  Cn × Cn , where zi =

98

Chapter 2. Algebraic Integral Geometry

√ √ xi + −1 yi and ζi = ξi + −1 ηi . The natural action of U (n) on T Cn corresponds to the diagonal action on Cn × Cn . Following Park [53], we consider the elements θ0 := θ1 := θ2 :=

n  i=1 n  i=1 n 

dξi ∧ dηi , (dxi ∧ dηi − dyi ∧ dξi ) , dxi ∧ dyi ,

i=1

in Λ2 (Cn ⊕ Cn )∗ . Thus θ2 is the pullback via the projection map T Cn → Cn of the K¨ahler form of Cn , and θ0 + θ1 + θ2 is the pullback of the K¨ahler form under exponential map exp(z, ζ) := z + ζ. Together with the symplectic form

the n ω = i=1 (dxi ∧ dξi + dyi ∧ dηi ), the θi generate the algebra of all U (n)-invariant elements in Λ∗ (Cn × Cn ). For positive integers k and q with max{0, k − n} ≤ q ≤ k2 ≤ n, we now set θk,q := cn,k,q θ0n+q−k ∧ θ1k−2q ∧ θ2q ∈ Λ2n (Cn × Cn ) for cn,k,q to be specified below, and put



μk,q (K) :=

θk,q .

(2.5.11)

N1 (K)

It will be useful to understand the Klain functions associated to the elements of ValU (n) . Clearly, they are invariant under the action of U (n) on the (real) Grassmannian of Cn . Tasaki [59, 60] classified the U (n) orbits of the Grk (Cn ) by ahler angle Θ(E) := (0 ≤ θ1 (E) ≤ · · · ≤ defining for E ∈ Grk (Cn ) the multiple K¨ θp (E) ≤ π2 ), p :=  k2 , via the condition that there exists an orthonormal basis ahler form of Cn α1 , . . . , αk of the dual space E ∗ such that the restriction of the K¨ to E is k 2  cos θi α2i−1 ∧ α2i . i=1

This is a complete invariant of the orbit. If k > n, then   ⊥ Θ(E) = 0, . . . , 0, Θ(E ) . 0 12 3 k−n

We put Grk,q for the orbit of all E = E k,q that may be expressed as the orthogonal direct sum of a q-dimensional complex subspace and a complementary (k − 2q)-dimensional subspace isotropic with respect to the K¨ahler form, i.e.,   π π k,q Θ(E ) = 0, . . . , 0, , . . . , . 0 12 3 02 12 23 q p−q

2.5. Hermitian integral geometry

99

In particular, Gr2p,p is the Grassmannian of p-dimensional complex subspaces and Grn,0 (Cn ) is the Lagrangian Grassmannian. It is not hard to prove the following: U (n)

Lemma 2.5.5. The Klain function map gives a linear isomorphism between Valk and the vector space of symmetric polynomials in cos2 Θ(E) := (cos2 θ1 (E), . . . , cos2 θp (E))

for k ≤ n, and in cos2 Θ(E ⊥ ) if k > n. The Hermitian intrinsic volumes are characterized by the condition Klμk,q (E k,l ) = δlq

(2.5.12)

and the Alesker–Fourier transform acts on them by μ4 k,q = μ2n−k,n−k+q .

(2.5.13)

The valuation μn,0 ∈ ValU (n) is called the Kazarnovskii pseudo-volume. Note that the restriction of μn,0 to ValU (n−1) (Cn−1 ) is zero: there are no isotropic planes of dimension n in Cn−1 . Since by Theorem 2.5.3 the kernel of the restriction map is spanned by the polynomial fn (s, t), we obtain Lemma 2.5.6. μk,0 = cfk for some constant c. The following is elementary. Lemma 2.5.7. Put Σp for the vector space spanned by the elementary symmetric functions in x1 , . . . , xp . Let v0 := (0, . . . , 0), v1 := (0, . . . , 0, 1), . . . , vp := (1, . . . , 1) denote the vertices of the simplex Δp := {0 ≤ x1 ≤ x2 ≤ · · · ≤ xp ≤ 1} and denote by g0 , . . . , gp the basis of Σp determined by the conditions gi (vj ) = δji . Then the restriction of g ∈ Σp to Δp is nonnegative if and only if g ∈ g0 , . . . , gp + . From this we deduce at once U (n)

Proposition 2.5.8. The positive cone P ∩ Valk

is equal to μk,0 , . . . , μk,p + . U (n)

It turns out that the Crofton-positive cone CP ∩ Valk is the cone on the valuations νk,q , where the invariant probability measure on Grk,q is a Crofton measure for νk,q . Furthermore, using the explicit representation for the Hermitian intrinsic volumes in terms of differential forms, it is possible to compute their first variation curvature measures explicitly. Solving the inequalities δμk,q ≥ 0 and applying Theorem 2.4.9, the monotone cone M ∩ ValU (n) has been determined in [25]. All of these cones are polyhedral. However, in contrast to the SO(n) case described at the end of Section 2.4.6, for n ≥ 2 the three cones are pairwise distinct, and for n ≥ 4 the monotone cone is not simplicial.

100

Chapter 2. Algebraic Integral Geometry

2.5.3 Tasaki valuations and a mysterious duality The Tasaki valuations τk,q for q ≤

k 2

are defined by the relations   Klτk,q (E) = σp,q cos2 Θ(E) ,

where σp,q is the qth elementary symmetric function in p := k/2 variables. U (n) If k ≤ n, then the Tasaki valuations τk,q constitute a basis for Valk ; on the other hand, if k > n, then the first K¨ ahler angle is identically zero and so the Tasaki valuations are linearly dependent in this range. It is therefore natural to consider the basis of ValU (n) consisting of the τk,q for k ≤ n, together with their (n) ). Fourier transforms (the Fourier transform acts trivially in middle degree ValU n We may now write kU (n) (χ) =

 

p,q≤ n 2

Tnn



τ ⊗ τn,q + pq n,p

n−1 



Tkn

 pq

(τk,p ⊗ τ4 4 k,q + τ k,q ⊗ τk,p ) ,

k=0 p,q≤min( k ,n− k ) 2 2

(2.5.14) where the Tkn are the Tasaki matrices, symmetric matrices of size min( k2 ,  2n−k ) 2 = τ ). In these terms, the unitary principal kinematic formula (note that τ4 n,p n,p exhibits a mysterious additional symmetry. Theorem 2.5.9. If k = 2l ≤ n is even, then (Tkn )i,j = (Tkn )l−i,l−j ,

0 ≤ i, j ≤ l.

(2.5.15)

There are two main ingredients for the proof, arising from two different algebraic representations of the Tasaki valuations and their Fourier transforms. The first has to do with how the Fourier transform behaves with regard to the representation of the Tasaki valuations in terms of elementary symmetric functions. For simplicity, we restrict to valuations of even degree. Let Σp denote the vector space of elementary symmetric functions σp,q in p variables. As noted above, if dim E = 2p < n, then the K¨ahler angle Θ(E ⊥ ) is just Θ(E) preceded by n − 2p zeroes, hence cos2 Θ(E ⊥ ) is just cos2 Θ(E) preceded by that many ones. In order to express τ4 2p,q as a linear combination of Tasaki valuations τ2n−2p,r in complementary degree, it is enough to express the qth elementary symmetric function σp,q (x1 , . . . , xp ) as a linear combination of the σn−p,r (1, . . . , 1, x1 , . . . , xp ). However, it is simpler to consider the map in the other direction, which corresponds to the map r : Σn−p → Σp given by   (2.5.16) r : f −→ fˆ = f 1, . . . , 1, x1 , . . . , xp . The second ingredient is the remarkable fact that the Tasaki valuations are monomials under a certain change of variable. Proposition 2.5.10. Put u := 4s − t2 . Then τk,q =

πk tk−2q uq . ωk (k − 2q)!(2q)!

(2.5.17)

2.5. Hermitian integral geometry

101

√ It is useful to introduce the formal complex variable z := t + −1v, where v is formally real and v 2 = u. Then      v2 t2 fk = log 1 + s + t = log 1 + t + + 4 4 k   5 5 z 2 z 66 , = log 1 + = 2 Re log 1 + 2 2 whence fk = c Re(z k ). U (n)

(2.5.18) U (n)

We introduce the linear involution ι : Val2∗ → Val2∗ on the subspace of valuations of even degree, determined by its action on Tasaki valuations: ι(τ2l,q ) := τ2l,l−q . Expressing these valuations as real polynomials of even degree in the real and imaginary parts of the formal complex variable z, this amounts to interchanging the real and imaginary parts t and v, i.e., √ z ). ι(p(z)) = p( −1¯ This is clearly a formal algebra automorphism of Reven [t, u]. Furthermore, it deU (n) scends to an automorphism of the quotient Val2∗ : to see this it is enough to check that the action of ι on the basic relations is given by ι(f2k ) = (−1)k f2k ,   4k f2k + tf2k−1 . ι(tf2k−1 ) = (−1)k+1 2k − 1 This is easily accomplished using the expression (2.5.18). Furthermore, ι commutes with the Fourier transform. To see this we return to the model spaces Σp of elementary symmetric functions. Here the map ι corresponds to σp,q → σp,p−q . The assertion thus reduces to the claim that for m = n − p ≥ p the diagram ι / Σm Σm r

 Σp

r

ι

 / Σp

commutes. It is enough to prove this for m = p + 1, in which case r(σp+1,i ) = σp,i + σp,i−1 . Hence, for i = 0, . . . , p + 1, ι ◦ r(σp+1,i ) = ι(σp,i + σp,i−1 ) = σp,p−i + σp,p−i+1 = r(σp+1,p−i+1 ) = r ◦ ι(σp+1,i ).

102

Chapter 2. Algebraic Integral Geometry U (n)

Proof of Theorem 2.5.9. Using the additional fact that ι acts trivially on Val2n , we compute  τ2p,i · τ4 2p,j = τ2p,i · (ιτ2p,p−j ) = τ2p,i · ι(τ 2p,p−j )   = ι τ2p,i · ι(τ 2p,p−j ) = ι(τ2p,i ) · τ 2p,p−j = τ2p,p−i · τ 2p,p−j . With Theorem 2.3.16, this implies the result.

2.5.4



Determination of the kinematic operator of (Cn , U (n))

The key to carrying out actual computations with the U (n)-invariant valuations is the correspondence between their algebraic representations in terms of t, s, u and the more geometric expressions in terms of the Hermitian intrinsic volumes and the Tasaki valuations. This is possible due mainly to two facts. The first is Lemma 2.5.6, establishing this correspondence in a crucial special case. The second is the explicit calculation, using Corollary 2.3.6, of the multiplication operator L from Theorem 2.3.2 (7) in terms of the Hermitian intrinsic volumes. By part (3) of Proposition 2.3.10, this gives in explicit terms the structure of U (n) Val as an sl(2)-module, and we can apply the well-established theory of such representations (cf., e.g., [44]). In particular, we find the primitive elements π2r,r ∈ U (n) Val2r for 2r ≤ n, characterized by the condition Λπ2r,r = L2n−2r+1 π2r,r = 0, that generate ValU (n) as an sl(2, R)-module in the sense that the valuations π2r+k,r := cLk π2r,r for 2r ≤ n and k ≤ 2n − 2r constitute another basis of ValU (n) . It follows at once from the definition that the Poincar´e pairing satisfies   πk,p , πl,q = 0 unless k + l = 2n and p = q. In other words, the Poincar´e pairing on ValU (n) is antidiagonal with respect to this basis, and moreover the precise forms of the formulas that led us to this point permit us to calculate the value of the product in the nontrivial cases. Thus Theorem 2.3.16 (and the remarks following it) yields the value of kU (n) (χ) in these terms. Unwinding the construction of the primitive basis, it is straightforward also to express kU (n) (χ) in terms of the Tasaki basis, although the coefficients are expressed as singly-indexed sums not in closed form. Since by (2.5.17) it is easy to compute the product of two Tasaki valuations, the multiplicative property of the kinematic operator allows us now to compute all of the kU (n) (τk,p ) and thereby also the kU (n) (μk,p ).

2.5. Hermitian integral geometry

103

With the transfer principle and (2.3.24) of Theorem 2.3.16 we can also compute precise first-order formulas in complex space forms, as well as additive kinematic formulas in Cn (note that the first-order formulas in CP n may be regarded as a generalization of B´ezout’s theorem from elementary algebraic geometry). For example, we may compute the expected value of the length of the curve given by the intersection of a real 4-fold and a real 5-fold in CP 4 . Theorem 2.5.11. Let M 4 , N 5 ⊂ CP 4 be real C 1 submanifolds of dimension 4 ahler angles of the tangent plane to M at and 5, respectively. Let θ1 , θ2 be the K¨ a general point x and ψ the K¨ ahler angle of the orthogonal complement to the tangent plane to N at y. Let dg denote the invariant probability measure on U (5). Then  length(M ∩ gN ) dg U (5)

7  1 × 30 vol (M ) vol (N ) − 6 vol (M ) cos2 ψ dy = 4 5 4 5π 4 N  −3 (cos2 θ1 + cos2 θ2 ) dx · vol5 (N ) M 8   2 2 2 +7 (cos θ1 + cos θ2 ) dx · cos ψ dy . M

N

The companion result is the following additive kinematic formula for the average 7-dimensional volume of the Minkowski sum of two convex subsets in C4 of dimensions 3 and 4, respectively. Theorem 2.5.12. Let E ∈ Gr4 (C4 ) and F ∈ Gr3 (C4 ). Let θ1 , θ2 be the K¨ ahler angles of E and ψ the K¨ ahler angle of F . Let dg be the invariant probability measure on U (4). If A ∈ K(E) and B ∈ K(F ), then  1 vol4 (A) vol3 (B)× vol7 (A + gB) dg = 120 U (4)  30 − 6 cos2 ψ − 3(cos2 θ1 + cos2 θ2 ) + 7 cos2 ψ(cos2 θ1 + cos2 θ2 ) .

2.5.5 Integral geometry of complex space forms In view of Theorem 2.4.13, the first-order kinematic formulas are identical in all three complex space forms, so the results above give the first-order formulas also for the curved complex space forms CP n and CH n . The full kinematic formulas, both for valuations and for curvature measures, have very recently been worked out in [26]. These results require additional ideas and techniques, which we cannot summarize adequately here. However, the quasi-combinatorial approach of Section 2.5.1 allows us to express some of these results in purely algebraic form. The calculations of Gray (specifically Corollary 6.25 and Theorem 7.7 of [43]) together with the identities

104

Chapter 2. Algebraic Integral Geometry

(2.3.27) and Section 2.1.5 above, show that the Lipschitz–Killing curvatures of the complex projective space with constant holomorphic sectional curvature 4 are given by  l  k+1  , if l even, l l l k 2 2 +1 (2.5.19) t (CP ) = 0, if l odd. Thus if we recall the definition (2.5.6) of s¯ ∈ V2C,1,n , then  k l n s¯ t (CP ) = χ( · ∩ Q) dQ GrC n−k

= tl (CP n−k )  l n−k+1 =

l 2

l 2 +1

(2.5.20) ,

0,

if l even, if l odd.

By the transfer principle (Theorem 2.4.13), the monomials in s¯ and t corresponding to the first Alesker basis for ValU (n) constitute a basis of V C,1,n , and there are no relations in filtrations ≤ n, where we put V C,λ,n for the algebra of invariant valuations on the complex space form of holomorphic sectional curvature 4λ. As in the proof of Theorem 2.5.3, by Poincar´e duality the ideal I of all relations between s¯ and t consists precisely of the polynomials f (¯ s, t) such that s, t))(CP n ) = 0. (¯ sk tl · f (¯ Using the evaluations (2.5.20) this condition amounts to a family of identities among the coefficients of f , which are in fact equivalent to the structural results of [26]. These identities are very complicated, to the point that it seems implausible to prove them directly in a satisfying way —even the Wilf–Zeilberger approach of [54] yields a certificate which is many pages long. On the contrary, at this point it seems better to view the identities as consequences of [26]. Remarkably, it turns out that for fixed n the algebras V C,λ,n are all isomorphic, and in fact there are a number of interesting explicit isomorphisms among them. One of the most attractive is the following. Theorem 2.5.13 (Bernig–Fu–Solanes [26]).   t V C,λ,n  R[[s, t]]/ fn+1 s¯, + 1+

 λt2 4

 , fn+2 s¯, +



t 1+

λt2 4

,

where the fi are as in Theorem 2.5.3. Chronologically, the first conjecture about the structure of the algebras V C,λ,n was the following, which was arrived at through a combination of luck and consultation with the Online Encyclopedia of Integer Sequences.

2.6. Concluding remarks

105

Conjecture 2.5.14. Let s¯, t and λ be variables of formal degrees 2, 1 and −2, s, t, λ) by respectively. Define the formal series f¯k (¯    s, t, λ)xk = log 1 + s¯x2 + tx + λx−2 + 3λ2 x−4 + 13λ3 x−6 + · · · (2.5.21) f¯k (¯ 8   74n + 1   4n + 1 λn x−2n . −9 = log 1 + s¯x2 + tx + n−1 n+1 (2.5.22) Then f¯i (¯ s, t, λ) = 0 in V C,λ,n for all i > n. Modulo filtration n+3, Conjecture 2.5.14 is independent of λ and is therefore true by Theorem 2.5.3. In particular, the conjecture holds through dimension n = 2. It is possible to reduce the full conjecture to a sequence of identities similar to (2.5.8). For example, the validity of the conjecture modulo filtration n + 5 is equivalent to the family of identities  n+3 2 



(−1)j+1

j=0

   2n − 2k − 2j n−j+4 1 n−k−j n − j + 4 1, j, n − 2j + 3    n − k − i + 1 n − i + 1 2n − 2k − 2i − 2 = 0, n−k−i−1 i n−i+1 i=0 9 : n−3 k = 0, . . . , . 2  n+1 2 

+



(−1)i

Gessel [42] has proved this to be true by a method similar to (but more complicated than) the proof of (2.5.8). Hence we know that the conjecture is true modulo filtration n + 5, and through dimension n = 4. In principle one could continue in this way, but the numerical identities that arise become unreasonably complicated. Nevertheless, using this approach we have confirmed by direct machine calculation that Conjecture 2.5.14 is true through dimension n = 16, which involves the first seven terms of the λ series, and Bernig has reported that it remains true for still larger values of n. Completely separate from any thoughts of integral geometry, a combinatorial description of the coefficients appearing in (2.5.22) has been given by F. Chapoton [30], who also proved that the λ series there defines an algebraic function g(λ) satisfying   4  g = f 1 − f − f2 , f = λ 1 + f . This last fact was also observed independently by Gessel.

2.6

Concluding remarks

The theory outlined above gives rise to many open problems. Here are a few general directions that should prove fruitful in the next stage of development.

106

Chapter 2. Algebraic Integral Geometry

(1) Further exploration of the integral geometry of (Cn , U (n)). The sizable literature that surrounds the classical SO(n) kinematic formula, seemingly out of all proportion with the extreme simplicity of the underlying algebra, suggests that the identities of [25] may represent only the tip of the iceberg, and unitary integral geometry may hold many more wonderful surprises even in the relatively well-understood Euclidean case. In my opinion it will likely be fruitful simply to play with the algebra and see what comes out. With luck this approach may lead to better structural understanding of unitary integral geometry —for example, this is precisely what happened in the development of the duality theorem (Theorem 2.5.9). For instance, is there a better combinatorial model for the kinematic formulas, perhaps something like the devices occurring in Schubert calculus? Does the combinatorial construction of [30] shed light on Conjecture 2.5.14? (2) Structure of the array {V C,λ,n }λ,n . In each dimension n, the spaces of curvature measures may be thought of as a single vector space C = Cn , equipped with a coproduct k˜ = k˜n : C → C ⊗ C, independent of curvature. The coalgebra V C,λ,n of invariant valuations in the complex space form of holomorphic sectional curvature λ is then a quotient of C by a subspace Kλ . Furthermore, C is a module over each V C,λ,n . It turns out ([26]) that the actions of the various V C,λ,n on C all commute with each other. This phenomenon is actually very general, and applies also to the other curvature-dependent families of isotropic spaces, namely the real space forms, the quaternionic space forms, and the Cayley space forms. In the real case they are implicit in the discussion of Section 2.4.6. (3) Quaternionic integral geometry. Of the isotropic spaces of whose integral geometry we remain ignorant, the natural isotropic structures on the quaternionic space forms seems most central and mysterious. By a tour de force calculation, Bernig [20] has determined the Betti numbers of the algebras ValG (Hn ) associated to the quaternionic vector space Hn and the natural isotropic actions of G = Sp(n), Sp(n) × U (1) and Sp(n) × Sp(1). The last group is the appropriate one for the associated family of quaternionic space forms. But beyond this nothing is known about the underlying algebra. Then there are the octonian (Cayley) spaces: the Euclidean version is the isotropic action of Spin(9) on R16 . (4) The Weyl principle gives rise to the Lipschitz–Killing algebra and hence has central importance. Nevertheless, it remains poorly understood. For example, the proof above of Theorem 2.1.10 has the form of a numerical accident. It appears that the U (n)-invariant curvature measures may also restrict on submanifolds to some kind of intrinsic quantities, perhaps invariants of the combination of the induced Riemannian metric and the restriction of the

2.6. Concluding remarks

107

K¨ahler form. The same seems to be true for the Holmes–Thompson valuations μF i of Section 2.3.4, although, curiously, these valuations seem not to correspond to intrinsic curvature measures. Furthermore, in conjunction with the analytic questions that we have so far avoided mentioning, the Weyl principle hints at a potentially vast generalization of Riemannian geometry to singular spaces. For example, it can be proved by ad hoc means that the Federer curvature measures of a subanalytic subset of Rn are intrinsic (cf. [16, 22, 29, 37, 38, 39]). We believe that the Weyl principle should remain true for any subspace of Rn that admits a normal cycle, despite the possible severity of their singularities.

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birkhauser-science.com Advanced Courses in Mathematics – CRM Barcelona (ACM) Edited by Carles Casacuberta, Universitat de Barcelona, Spain Since 1995 the Centre de Recerca Matemàtica (CRM) has organised a number of Advanced Courses at the post-doctoral or advanced graduate level on forefront research topics in Barcelona. The books in this series contain revised and expanded versions of the material presented by the authors in their lectures.

 Böckle, G. / Burns, D. / Goss, D. / Thakur, D. / Trihan, F. / Ulmer, D., Arithmetic Geometry over Global Function Fields (2014). ISBN 978-3-0348-0852-1

This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009–2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.  Cruz-Uribe, D. / Fiorenza, A. / Ruzhansky, M. / Wirth, J., Variable Lebesgue Spaces and Hyperbolic Systems (2014). ISBN 978-3-0348-0839-2

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations

and systems with time-dependent coefficients. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.  Berger, L. / Böckle, G. / Dembélé, L. / Dimitrov, M. / Dokchitser, T. / Voight, J., Elliptic Curves, Hilbert Modular Forms and Galois Deformations (2013). ISBN 978-3-0348-0617-6

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009–2010 academic year. They provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory, offer a comprehensive course on Galois deformation theory, present the basics of the arithmetic theory of Hilbert modular forms and varieties, describe methods for performing explicit computations in spaces of Hilbert modular forms, and describe the proof of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group.  Cominetti, R. / Facchinei, F. / Lasserre, J.B., Modern Optimization Modelling Techniques (2012). ISBN 978-3-0348-0290-1  Caffarelli, L.A. / Golse, F. / Guo, Y. / Kenig, C.E. / Vasseur, A., Nonlinear Partial Differential Equations (2012). ISBN 978-3-0348-0190-4  Moerdijk, I. / Toën, B., Simplicial Methods for Operads and Algebraic Geometry (2010). ISBN 978-3-0348-0051-8  Ritoré, M. / Sinestrari, C., Mean Curvature Flow and Isoperimetric Inequalities (2010). ISBN 978-3-0346-0212-9

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