Radon transforms and the rigidity of the Grassmannians / Jacques Gasqui and Hubert Goldschmidt.

Format:

Book

Author/Creator:

Gasqui, Jacques.

Contributor:

Goldschmidt, Hubert, 1942-

Series:

Annals of mathematics studies ; no. 156.

Annals of mathematics studies ; no. 156

Language:

English

Subjects (All):

Radon transforms.

Grassmann manifolds.

Physical Description:

1 online resource (385 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, 2004.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

TABLE OF CONTENTS

INTRODUCTION

Chapter I. Symmetric Spaces and Einstein Manifolds

Chapter II. Radon Transforms on Symmetric Spaces

Chapter III. Symmetric Spaces of Rank One

Chapter IV. The Real Grassmannians

Chapter V. The Complex Quadric

Chapter VI. The Rigidity of the Complex Quadric

Chapter VII. The Rigidity of the Real Grassmannians

Chapter VIII. The Complex Grassmannians

Chapter IX. The Rigidity of the Complex Grassmannians

Chapter X. Products of Symmetric Spaces

References

Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [357]-361) and index.

ISBN:

9786612158988

9781282158986

1282158988

9781400826179

1400826179

OCLC:

437268713