Geometric Analysis on Symmetric Spaces (Mathematical Surveys and Monographs)

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. Algebraic curves and compact Riemann surfaces comprise the most developed and arguably the most beautiful portion of algebraic geometry. However, the majority of books written on the subject discuss algebraic curves and compact Riemann surfaces separately, as parts of distinct general theories. Most texts and university courses on curve theory generally conclude with the Riemann-Roch theorem, despite the fact that this theorem is the gateway to some of the most fascinating results in the theory of algebraic curves. This book is based on a six-week series of lectures presented by the author to third- and fourth-year undergraduates and graduate students at Beijing University in 1982. The lectures began with minimal technical requirements (a working knowledge of elementary complex function theory and algebra together with some exposure to topology of compact surfaces) and proceeded directly to the Riemann-Roch and Abel theorems. This book differs from a number of recent books on this subject in that it combines analytic and geometric methods at the outset, so that the reader can grasp the basic results of the subject. Although such modern techniques of sheaf theory, cohomology, and commutative algebra are not covered here, the book provides a solid foundation to proceed to more advanced texts in general algebraic geometry, complex manifolds, and Riemann surfaces, as well as algebraic curves. Containing numerous exercises and two exams, this book would make an excellent introductory text. "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."--Jacket