Radon Transforms and the Rigidity of the Grassmannians (Annals of Mathematics Studies, 156) - Hardcover

Gasqui, Jacques; Goldschmidt, Hubert

Hardcover

ISBN 10: 0691118981 ISBN 13: 9780691118987

Publisher: Princeton University Press, 2004

Synopsis

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.

About this title

About the Author

Jacques Gasqui is Professor of Mathematics at Institut Fourier, Université de Grenoble I. Hubert Goldschmidt is Visiting Professor of Mathematics at Columbia University and Professeur des Universités in France.

Edition details

• Publisher: Princeton University Press
• Publication date: 2004
• Language: English
• ISBN 10: 0691118981
• ISBN 13: 9780691118987
• Binding: Hardcover
• Edition number: 1
• Number of pages: 384