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Hardcover ISBN:  9780821828489 
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Hardcover ISBN:  9780821828489 
Product Code:  GSM/34 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470411466 
Product Code:  GSM/34.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821828489 
eBook ISBN:  9781470411466 
Product Code:  GSM/34.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 34; 2001; 641 ppMSC: Primary 22; 32; 43; 53
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 thoughtout 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, selfcontained 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.
ReadershipGraduate students and research mathematicians interested in differential geometry, Lie groups, and symmetric spaces.

Table of Contents

Chapters

Chapter I. Elementary differential geometry

Chapter II. Lie groups and Lie algebras

Chapter III. 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


Additional Material

Reviews

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 nonspecialist, 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 wellwritten 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


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 thoughtout 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, selfcontained 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.
Graduate students and research mathematicians interested in differential geometry, Lie groups, and symmetric spaces.

Chapters

Chapter I. Elementary differential geometry

Chapter II. Lie groups and Lie algebras

Chapter III. 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

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 nonspecialist, 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 wellwritten 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