# Differential Geometry, Lie Groups, and Symmetric Spaces

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PREFACE
The present book is intended as a textbook and a reference work on the three topics in the title. Together with a volume in progress on “Groups and Geometric Analysis” it supersedes my “Differential Geometry and Symmetric Spaces,” published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.

This first volume is an extensive revision of a part of “Differential Geometry and Symmetric Spaces.” Apart from numerous minor changes the following material has been added:

Chapter I, §15; Chapter II, §7-§8; Chapter III, §8; Chapter VII, §§7, 10, 11 and most of §2 and of §8; Chapter VIII, part of §7; all of Chapter IX and most of Chapter X. Many new exercises have been added, and solutions to the old and new exercises are now included and placed toward the end of the book.

The book begins with a general self-contained exposition of differential and Riemannian geometry, discussing affine connections, exponential mapping, geodesics, and curvature. Chapter II develops the basic theory of Lie groups and Lie algebras, homogeneous spaces, the adjoint group, etc. The Lie groups that are locally isomorphic to products of simple groups are called semisimple. These Lie groups have an extremely rich structure theory which at an early stage led to their complete classification, and which presumably accounts for their pervasive influence on present-day mathematics. Chapter III deals with their preliminary structure theory with emphasis on compact real forms.

Chapter IV is an introductory geometric study of symmetric spaces. According to its original definition, a symmetric space is a Riemannian manifold whose curvature tensor is invariant under all parallel translations. The theory of symmetric spaces was initiated by É. Cartan in 1926 and was vigorously developed by him in the late 1920s. By their definition, symmetric spaces form a special topic in Riemannian geom-

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