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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.

Radon transforms. --- Grassmann manifolds. --- Grassmannians --- Transforms, Radon --- Differential topology --- Manifolds (Mathematics) --- Integral geometry --- Integral transforms --- Adjoint. --- Automorphism. --- Cartan decomposition. --- Cartan subalgebra. --- Casimir element. --- Closed geodesic. --- Cohomology. --- Commutative property. --- Complex manifold. --- Complex number. --- Complex projective plane. --- Complex projective space. --- Complex vector bundle. --- Complexification. --- Computation. --- Constant curvature. --- Coset. --- Covering space. --- Curvature. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dot product. --- Eigenvalues and eigenvectors. --- Einstein manifold. --- Elliptic operator. --- Endomorphism. --- Equivalence class. --- Even and odd functions. --- Exactness. --- Existential quantification. --- G-module. --- Geometry. --- Grassmannian. --- Harmonic analysis. --- Hermitian symmetric space. --- Hodge dual. --- Homogeneous space. --- Identity element. --- Implicit function. --- Injective function. --- Integer. --- Integral. --- Isometry. --- Killing form. --- Killing vector field. --- Lemma (mathematics). --- Lie algebra. --- Lie derivative. --- Line bundle. --- Mathematical induction. --- Morphism. --- Open set. --- Orthogonal complement. --- Orthonormal basis. --- Orthonormality. --- Parity (mathematics). --- Partial differential equation. --- Projection (linear algebra). --- Projective space. --- Quadric. --- Quaternionic projective space. --- Quotient space (topology). --- Radon transform. --- Real number. --- Real projective plane. --- Real projective space. --- Real structure. --- Remainder. --- Restriction (mathematics). --- Riemann curvature tensor. --- Riemann sphere. --- Riemannian manifold. --- Rigidity (mathematics). --- Scalar curvature. --- Second fundamental form. --- Simple Lie group. --- Standard basis. --- Stokes' theorem. --- Subgroup. --- Submanifold. --- Symmetric space. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor. --- Theorem. --- Topological group. --- Torus. --- Unit vector. --- Unitary group. --- Vector bundle. --- Vector field. --- Vector space. --- X-ray transform. --- Zero of a function.