Differential Geometry, Lie Groups, and Symmetric Spaces

Sigurdur Helgason

• Publisher: American Mathematical Society
• Year: 2001
• Topic: Mathematics
• Subtopic: Mathematics General
• eBook ISBN:
  9781470411466
• Index: Mathematics

A great book … a necessary item in any mathematical library. —S. S. Chern, University of California A brilliant book: rigorous, tightly organized, and covering a vast amount of good mathematics. —Barrett O'Neill, University of California This is obviously a very valuable and well thought-out book on an important subject. —André Weil, Institute for Advanced Study The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been—and continues to be—the standard source for this material. Helgason begins with a concise, self-contained introduction to differential geometry. Next is a careful treatment of the foundations of the theory of Lie groups, presented in a manner that since 1962 has served as a model to a number of subsequent authors. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing–Cartan classification of simple Lie algebras over $\mathbb{C}$ and Cartan's classification of simple Lie algebras over $\mathbb{R}$, following a method of Victor Kac. The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All of the problems have either solutions or substantial hints, found at the back of the book. For this edition, the author has made corrections and added helpful notes and useful references. Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.

Table of contents

• Front Cover
• CONTENTS
• PREFACE
• Preface to the 2001 Printing
• SUGGESTIONS TO THE READER
• Sequel to the Present Volume
• GROUPS AND GEOMETRIC ANALYSIS
• GEOMETRIC ANALYSIS ON SYMMETRIC SPACES
• CHAPTER I ELEMENTARY DIFFERENTIAL GEOMETRY
• CHAPTER II LIE GROUPS AND LIE ALGEBRAS
• CHAPTER Ill STRUCTURE OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER IV SYMMETRIC SPACES
• CHAPTER V DECOMPOSITION OF SYMMETRIC SPACES
• CHAPTER VI SYMMETRIC SPACES OF THE NONCOMPACT TYPE
• CHAPTER VII SYMMETRIC SPACES OF THE COMPACT TYPE
• CHAPTER VIII HERMITIAN SYMMETRIC SPACES
• CHAPTER IX STRUCTURE OF SEMISIMPLE LIE GROUPS
• CHAPTER X THE CLASSIFICATION OF SIMPLE LIE ALGEBRAS AND OF SYMMETRIC SPACES
• SOLUTIONS TO EXERCISES
• SOME DETAILS
• Supplementary Notes
• Errata
• BIBLIOGRAPHY
• LIST OF NOTATIONAL CONVENTIONS
• SYMBOLS FREQUENTLY USED
• INDEX
• Reviews for the First Edition
• Back Cover