Radon transforms and the rigidity of the grassmannians

385 Pages·2004·1.57 MB·english

by  Jacques Gasqui

Annals of Mathematics Studies
Number 156
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Radon Transforms and the Rigidity
of the Grassmannians
JACQUES GASQUI
AND
HUBERT GOLDSCHMIDT
PRINCETON UNIVERSITY PRESS
Princeton and Oxford
2004
Copyright (cid:2)c 2004 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place,
Woodstock, Oxfordshire OX20 1SY
All Rights Reserved
Library of Congress Control Number 2003114656
ISBN: 0-691-11898-1
0-691-11899-X (paper)
British Library Cataloging-in-Publication Data is available
The publisher would like to acknowledge the authors of this
volume for providing the camera-ready copy from which this
book was printed
Printed on acid-free paper
pup.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
TABLE OF CONTENTS
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
CHAPTER I
Symmetric spaces and Einstein manifolds
1. Riemannian manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Einstein manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4. Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
CHAPTER II
Radon transforms on symmetric spaces
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
CHAPTER III
Symmetric spaces of rank one
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
CHAPTER IV
The real Grassmannians
1. The real Grassmannians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2. The Guillemin condition on the real Grassmannians. . . . . . . . . . 126
vi TABLEOFCONTENTS
CHAPTER V
The complex quadric
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 K¨ahler 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
CHAPTER VI
The rigidity of the complex quadric
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
CHAPTER VII
The rigidity of the real Grassmannians
1. The rigidity of the real Grassmannians . . . . . . . . . . . . . . . . . . . 244
2. The real Grassmannians G¯R . . . . . . . . . . . . . . . . . . . . . . . . . . 249
n,n
CHAPTER VIII
The complex Grassmannians
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
TABLEOFCONTENTS vii
5. ThecomplexGrassmanniansofranktwo. . . . . . . . . . . . . . . . . . 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 G¯C . . . . . . . . . . . . . . . . . . . . . . 303
n,n
CHAPTER IX
The rigidity of the complex Grassmannians
1. The rigidity of the complex Grassmannians. . . . . . . . . . . . . . . . 308
2. On the rigidity of the complex Grassmannians G¯C . . . . . . . . . . 313
n,n
3. The rigidity of the quaternionic Grassmannians. . . . . . . . . . . . . 323
CHAPTER X
Products of symmetric spaces
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 G¯R . . . . . . . . . . . . . . . . . . . . . . . . 340
2,2
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
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INTRODUCTION
This monograph is motivated by a fundamental rigidity problem in
Riemannian geometry: determine whether the metric of a given Rieman-
niansymmetricspaceofcompacttypecanbecharacterizedbymeansofthe
spectrum of its Laplacian. An infinitesimal isospectral deformation of the
metric of such a symmetric space belongs to the kernel of a certain Radon
transform defined in terms of integration over the flat totally geodesic tori
of dimension equal to the rank of the space. Here we study an infinitesi-
mal version of this spectral rigidity problem: determine all the symmetric
spaces of compact type for which this Radon transform is injective in an
appropriate sense. We shall both give examples of spaces which are not
infinitesimally rigid in this sense and prove that this Radon transform is
injective in the case of most Grassmannians.
Atpresent,itisonlyinthecaseofspacesofrankonethatinfinitesimal
rigidityinthissensegivesrisetoacharacterizationofthemetricbymeans
ofitsspectrum. Inthecaseofspacesofhigherrank,therearenoanalogues
ofthisphenomenonandtherelationshipbetweenthetworigidityproblems
is not yet elucidated. However, the existence of infinitesimal deformations
belonging to the kernel of the Radon transform might lead to non-trivial
isospectral deformations of the metric.
Here we also study another closely related rigidity question which
arises from the Blaschke problem: determine all the symmetric spaces for
which the X-ray transform for symmetric 2-forms, which consists in inte-
grating over all closed geodesics, is injective in an appropriate sense. In
the case of spaces of rank one, this problem coincides with the previous
Radon transform question. The methods used here for the study of these
two problems are similar in nature.
Let (X,g) be a Riemannian symmetric space of compact type. Con-
sider a family of Riemannian metrics {gt} on X, for |t| < ε, with g0 = g.
The family {g } is said to be an isospectral deformation of g if the spec-
t
trum of the Laplacian of the metric g is independent of t. We say that
t
the space (X,g) is infinitesimally spectrally rigid (i.e., spectrally rigid to
first-order) if, for every such isospectral deformation {g } of g, there is a
t
one-parameter family of diffeomorphisms {ϕ } of X such that g =ϕ∗g to
t t t
first-order in t at t = 0, or equivalently if the symmetric 2-form, which is
equal to the infinitesimal deformation d g of {g }, is a Lie derivative
dt t|t=0 t
of the metric g.