Radon Transforms and the Rigidity of the Grassmannians
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€127.99
A01=Hubert Goldschmidt
A01=Jacques Gasqui
Author_Hubert Goldschmidt
Author_Jacques Gasqui
Category=PB
Closed geodesic
Cohomology
Commutative property
Complex manifold
Complex number
Complex projective plane
Complex projective space
Complexification
Constant curvature
Coset
Curvature
Diffeomorphism
Differential form
Differential operator
Dimension (vector space)
Dot product
Eigenvalues and eigenvectors
Einstein manifold
Endomorphism
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equivalence class
Even and odd functions
Exactness
Existential quantification
G-module
Geometry
Grassmannian
Harmonic analysis
Hermitian symmetric space
Homogeneous space
Identity element
Injective function
Integer
Isometry
Killing form
Killing vector field
Lemma (mathematics)
Lie algebra
Lie derivative
Mathematical induction
Morphism
Orthogonal complement
Orthonormal basis
Orthonormality
Parity (mathematics)
Projection (linear algebra)
Projective space
Quadric
Radon transform
Real number
Real projective plane
Real projective space
Real structure
Restriction (mathematics)
Riemannian manifold
Rigidity (mathematics)
Simple Lie group
Subgroup
Submanifold
Symmetric space
Tangent bundle
Tangent space
Tensor
Theorem
Torus
Unit vector
Vector bundle
Vector field
Vector space
X-ray transform
Zero of a function
Product details
- ISBN 9780691118994
- Weight: 539g
- Dimensions: 152 x 235mm
- Publication Date: 25 Jan 2004
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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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.
Jacques Gasqui is Professor of Mathematics at Institut Fourier, Universite de Grenoble I. Hubert Goldschmidt is Visiting Professor of Mathematics at Columbia University and Professeur des Universites in France.
