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Radon transforms and the rigidity of the Grassmannians. (English) Zbl 1051.44003
Annals of Mathematics Studies 156. Princeton, NJ: Princeton University Press (ISBN 0-691-11899-X/pbk; 0-691-11898-1/hbk). xvii, 366 p. (2004).
The monograph is motivated by the fundamental rigidity problem in the Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of the spectrum of its Laplacian. The book consists of the following chapters: I. Symmetric spaces and Einstein manifolds, II. Radon transforms on symmetric spaces, III. Symmetric spaces of rank one, IV. The real Grassmannians, V. The complex quadric, VI. The rigidity of the complex quadric, VII. The rigidity of the real Grassmannians, VIII. The complex Grassmannians, IX. The rigidity of the complex Grassmannians, X. Product of symmetric spaces.
In the monograph the authors introduce new methods for studying the Guillemin rigidity of irreducible symmetric spaces of compact type.

44A12 Radon transform
53C65 Integral geometry
53C30 Differential geometry of homogeneous manifolds
44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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