Geometric analysis on symmetric spaces

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. "A great book! A major contribution to mathematical literature by presenting a lucid and well-organized account of a difficult subject ." -- S. S. Chern "Helgason’s monograph is a beautifully done piece of work and should prove to be extremely useful for several years to come, both in teaching and in research." -- D. Spencer "Although the topic of “symmetric spaces”was initiated nearly 40 years ago by Elie Cartan and has been cultivated by an ever increasing group of active mathematicians since, Helgason was the first to write a connected account about it, and it will undoubtedly be a ' must' for many years to come. It is written with unmatched lucidity, systematically, carefully, beautifully. Many familiar chains of reasoning have been rethought I by the author anew and the book gains in attractiveness with every page read." -- S. Bochner