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Differential Geometry, Lie Groups, and Symmetric Spaces
Sigurdur Helgason, Massachusetts Institute of Technology, Cambridge, MA

Graduate Studies in Mathematics
2001; 641 pp; hardcover
Volume: 34
ISBN-10: 0-8218-2848-7
ISBN-13: 978-0-8218-2848-9
List Price: US$80
Member Price: US$64
Order Code: GSM/34
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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.

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Graduate students and research mathematicians interested in differential geometry, Lie groups, and symmetric spaces.


"This book has been famous for many years and used by several generations of readers. It is important that the book has again become available for a general audience."

-- European Mathematical Society Newsletter

"One of the most important and excellent textbooks and a reference work about contemporary differential geometry ..."

-- Zentralblatt MATH

"Important improvements in the new edition of S. Helgason's book will turn it into a desk book for many following generations."

-- Mathematica Bohemica

From reviews for the First Edition:

"A great book ... a necessary item in any mathematical library."

-- S. S. Chern, University of California

"Written with unmatched lucidity, systematically, carefully, beautifully."

-- S. Bochner, Princeton University

"Helgason's monograph is a beautifully done piece of work and should be extremely useful for several years to come, both in teaching and in research."

-- D. Spencer, Princeton University

"A brilliant book: rigorous, tightly organized, and covering a vast amount of good mathematics."

-- Barrett O'Neill, University of California

"Renders a great service in permitting the non-specialist, with a minimum knowledge of differential geometry and Lie groups, an initiation to the theory of symmetrical spaces."

-- H. Cartan, Secretariat Mathématique, Paris

"The mathematical community has long been in need of a book on symmetric spaces. S. Helgason has admirably satisfied this need with his book, Differential Geometry and Symmetric Spaces. It is a remarkably well-written book ... a masterpiece of concise, lucid mathematical exposition ... it might be used as a textbook for "how to write mathematics"."

-- Louis Auslander

"[The author] will earn the gratitude of many generations of mathematicians for this skillful, tasteful, and highly efficient presentation. It will surely become a classic."

-- G. D. Mostow, Yale University

Table of Contents

  • Elementary differential geometry
  • Lie groups and Lie algebras
  • Structure of semisimple Lie algebras
  • Symmetric spaces
  • Decomposition of symmetric spaces
  • Symmetric spaces of the noncompact type
  • Symmetric spaces of the compact type
  • Hermitian symmetric spaces
  • Structure of semisimple Lie groups
  • 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
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