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.
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 CHAPTER V. The Complex Quadradic 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 Kahler 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 at 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 249 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 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 303 CHAPTER IX. The rigidity of the complex Grassmannians 1. The rigidity of the complex Grassmannians 308 2. On the rigidity of the complex Grassmannians 313 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 at symmetric spaces 334 3. Infinitesimal rigidity of products of symmetric spaces 338 4. The infinitesimal rigidity of G 340 References 357 Index 363