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Radon Transforms and the Rigidity of the Grassmannians (AM-156) : Annals of Mathematics Studies - Jacques Gasqui

Radon Transforms and the Rigidity of the Grassmannians (AM-156)

Annals of Mathematics Studies

Paperback

Published: 25th January 2004
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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.

Introductionp. ix
Symmetric spaces and Einstein manifolds
Riemannian manifoldsp. 1
Einstein manifoldsp. 15
Symmetric spacesp. 19
Complex manifoldsp. 27
Radon transforms on symmetric spaces
Outlinep. 32
Homogeneous vector bundles and harmonic analysisp. 32
The Guillemin and zero-energy conditionsp. 36
Radon transformsp. 41
Radon transforms and harmonic analysisp. 50
Lie algebrasp. 58
Irreducible symmetric spacesp. 59
Criteria for the rigidity of an irreducible symmetric spacep. 68
Symmetric spaces of rank one
Flat torip. 75
The projective spacesp. 83
The real projective spacep. 89
The complex projective spacep. 94
The rigidity of the complex projective spacep. 104
The other projective spacesp. 112
The real Grassmannians
The real Grassmanniansp. 114
The Guillemin condition on the real Grassmanniansp. 126
The Complex Quadradic
Outlinep. 134
The complex quadric viewed as a symmetric spacep. 134
The complex quadric viewed as a complex hypersurfacep. 138
Local Kauml;hler geometry of the complex quadricp. 146
The complex quadric and the real Grassmanniansp. 152
Totally geodesic surfaces and the infinitesimal orbit of the curvaturep. 159
Multiplicitiesp. 170
Vanishing results for symmetric formsp. 185
The complex quadric of dimension twop. 190
The rigidity of the complex quadric
Outlinep. 193
Total geodesic at tori of the complex quadricp. 194
Symmetric forms on the complex quadricp. 199
Computing integrals of symmetric formsp. 204
Computing integrals of odd symmetric formsp. 209
Bounds for the dimensions of spaces of symmetric formsp. 218
The complex quadric of dimension threep. 223
The rigidity of the complex quadricp. 229
Other proofs of the infinitesimal rigidity of the quadricp. 232
The complex quadric of dimension fourp. 234
Forms of degree onep. 237
The rigidity of the real Grassmannians
The rigidity of the real Grassmanniansp. 244
The real Grassmanniansp. 249
The complex Grassmannians
Outlinep. 257
The complex Grassmanniansp. 258
Highest weights of irreducible modules associated with the complex Grassmanniansp. 270
Functions and forms on the complex Grassmanniansp. 274
The complex Grassmannians of rank twop. 282
The Guillemin condition on the complex Grassmanniansp. 287
Integrals of forms on the complex Grassmanniansp. 293
Relations among forms on the complex Grassmanniansp. 300
The complex Grassmanniansp. 303
The rigidity of the complex Grassmannians
The rigidity of the complex Grassmanniansp. 308
On the rigidity of the complex Grassmanniansp. 313
The rigidity of the quaternionic Grassmanniansp. 323
Products of symmetric spaces
Guillemin rigidity and products of symmetric spacesp. 329
Conformally at symmetric spacesp. 334
Infinitesimal rigidity of products of symmetric spacesp. 338
The infinitesimal rigidity of G 340
Referencesp. 357
Indexp. 363
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691118994
ISBN-10: 069111899X
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 384
Published: 25th January 2004
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 2.54
Weight (kg): 0.54