Gives an interesting overview of the many problems in this field of mathematics--a welcome monograph for both students and specialists. -- G. Olafsson, Mathematical Reviews, 1997
Helgason sets a model of style and clarity that has not been matched since Enriques' "Geometria Projettiva". This is the kind of mathematics that will live forever. -- G-C. Rota, The Bulletin of Mathematical Books, Nov. 1994
The style is very fluent and pleasant, conducting the reader at a regular pace ... the present book will be a most valuable (and reasonably priced) reference for anyone interested in Radon transforms and analysis on semisimple Lie groups." -- --Bulletin of the AMS
This book gives the first systematic exposition of geometric analysis on Riemannian symmetric spaces and its relationship to the representation theory of Lie groups. The book starts with modern integral geometry for double fibrations and treats several examples in detail. After discussing the theory of Radon transforms and Fourier transforms on symmetric spaces, inversion formulas, and range theorems, Helgason examines applications to invariant differential equations on symmetric spaces, existence theorems, and explicit solution formulas, particularly potential theory and wave equations. The canonical multitemporal wave equation on a symmetric space is included. The book concludes with a chapter on eigenspace representations--that is, representations on solution spaces of invariant differential equations. Known for his high-quality expositions, Helgason received the 1988 Steele Prize for his earlier books Differential Geometry, Lie Groups and Symmetric Spaces and Groups and Geometric Analysis. Containing exercises (with solutions) and references to further results, this revised edition would be suitable for advanced graduate courses in modern integral geometry, analysis on Lie groups, and representation theory of Lie groups.