**Graduate Studies in Mathematics**

Volume: 34;
2001;
641 pp;
Hardcover

MSC: Primary 22; 32; 43; 53;

**Print ISBN: 978-0-8218-2848-9
Product Code: GSM/34**

List Price: $90.00

AMS Member Price: $72.00

MAA Member Price: $81.00

**Electronic ISBN: 978-1-4704-1146-6
Product Code: GSM/34.E**

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

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#### Supplemental Materials

# Differential Geometry, Lie Groups, and Symmetric Spaces

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*Sigurdur Helgason*

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.

#### Readership

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

#### Reviews & Endorsements

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

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

# Table of Contents

## Differential Geometry, Lie Groups, and Symmetric Spaces

- Cover Cover11
- Title page v6
- Copyright vi7
- Dedication page vii8
- Contents ix10
- Preface to original edition xiii14
- Preface to the 2001 printing xvii18
- Suggestions to the reader xix20
- Sequel to the present volume xxi22
- Groups and geometric analysis contents xxiii24
- Geometric analysis on symmetric spaces contents xxv26
- Elementary differential geometry 128
- Lie groups and Lie algebras 97124
- Structure of semisimple Lie algebras 155182
- Symmetric spaces 197224
- Decomposition of symmetric spaces 229256
- Symmetric spaces of the noncompact type 252279
- Symmetric spaces of the compact type 281308
- Hermitian symmetric spaces 352379
- Structure of semisimple Lie groups 401428
- The classification of simple Lie algebras and of symmetric spaces 438465
- Solutions to exercises 538565
- Some details 586613
- Bibliography 605632
- List of notational conventions 635662
- Symbols frequently used 638665
- Index 641668
- Reviews for the first edition 647674
- Back Cover Back Cover1675