Radon Transforms and the Rigidity of the Grassmannians

by Jacques Gasqui, Hubert Goldschmidt

Overview

This book provides the first unified examination of the relationship between Radon transforms on symmetric spaces of compact type and the infinitesimal versions of two fundamental rigidity problems in Riemannian geometry. Its primary focus is the spectral rigidity problem: Can the metric of a given Riemannian symmetric space of compact type be characterized by means of the spectrum of its Laplacian? It also addresses a question rooted in the Blaschke problem: Is a Riemannian metric on a projective space whose geodesics are all closed and of the same length isometric to the canonical metric?


The authors comprehensively treat the results concerning Radon transforms and the infinitesimal versions of these two problems. Their main result implies that most Grassmannians are spectrally rigid to the first order. This is particularly important, for there are still few isospectrality results for positively curved spaces and these are the first such results for symmetric spaces of compact type of rank >1. The authors exploit the theory of overdetermined partial differential equations and harmonic analysis on symmetric spaces to provide criteria for infinitesimal rigidity that apply to a large class of spaces.


A substantial amount of basic material about Riemannian geometry, symmetric spaces, and Radon transforms is included in a clear and elegant presentation that will be useful to researchers and advanced students in differential geometry.

Product Details

ISBN-13:	9780691118994
Publisher:	Princeton University Press
Publication date:	01/25/2004
Series:	Annals of Mathematics Studies , #156
Pages:	384
Product dimensions:	6.00(w) x 9.25(h) x (d)
Age Range:	18 Years

About the Author

Jacques Gasqui is Professor of Mathematics at Institut Fourier, Université de Grenoble I. Hubert Goldschmidt is Visiting Professor of Mathematics at Columbia University and Professeur des Universités in France.

Table of Contents

Contents

Introduction....................ix
1. Riemannian manifolds....................1
2. Einstein manifolds....................15
3. Symmetric spaces....................19
4. Complex manifolds....................27
1. Outline....................32
2. Homogeneous vector bundles and harmonic analysis....................32
3. The Guillemin and zero-energy conditions....................36
4. Radon transforms....................41
5. Radon transforms and harmonic analysis....................50
6. Lie algebras....................58
7. Irreducible symmetric spaces....................59
8. Criteria for the rigidity of an irreducible symmetric space....................68
1. Flat tori....................75
2. The projective spaces....................83
3. The real projective space....................89
4. The complex projective space....................94
5. The rigidity of the complex projective space....................104
6. The other projective spaces....................112
1. The real Grassmannians....................114
2. The Guillemin condition on the real Grassmannians....................126
1. Outline....................134
2. The complex quadric viewed as a symmetric space....................134
3. The complex quadric viewed as a complex hypersurface....................138
4. Local Kähler geometry of the complex quadric....................146
5. The complex quadric and the real Grassmannians....................152
6. Totally geodesic surfaces and the infinitesimal orbit of the curvature....................159
7. Multiplicities....................170
8. Vanishing results for symmetric forms....................185
9. The complex quadric of dimension two....................190
1. Outline....................193
2. Total geodesic flat tori of the complex quadric....................194
3. Symmetric forms on the complex quadric....................199
4. Computing integrals of symmetric forms....................204
5. Computing integrals of odd symmetric forms....................209
6. Bounds for the dimensions of spaces of symmetric forms....................218
7. The complex quadric of dimension three....................223
8. The rigidity of the complex quadric....................229
9. Other proofs of the infinitesimal rigidity of the quadric....................232
10. The complex quadric of dimension four....................234
11. Forms of degree one....................237
1. The rigidity of the real Grassmannians....................244
2. The real Grassmannians [[bar.G].sup.R.sub.n,n]....................249
1. Outline....................257
2. The complex Grassmannians....................258
3. Highest weights of irreducible modules associated with the complex Grassmannians....................270
4. Functions and forms on the complex Grassmannians....................274
5. The complex Grassmannians of rank two....................282
6. The Guillemin condition on the complex Grassmannians....................287
7. Integrals of forms on the complex Grassmannians....................293
8. Relations among forms on the complex Grassmannians....................300
9. The complex Grassmannians [[bar.G].sup.C.sub.n,n]....................303
1. The rigidity of the complex Grassmannians....................308
2. On the rigidity of the complex Grassmannians [[bar.G].sup.C.sub.n,n]....................313
3. The rigidity of the quaternionic Grassmannians....................323
1. Guillemin rigidity and products of symmetric spaces....................329
2. Conformally flat symmetric spaces....................334
3. Infinitesimal rigidity of products of symmetric spaces....................338
4. The infinitesimal rigidity of [[bar.G].sup.R.sub.2,2]....................340
References....................357
Index....................363