The author's earlier book [Differential geometry and symmetric spaces, Academic Press, New York, 1962; MR0145455] became one of the most important and most often cited works about contemporary differential geometry. The author now presents an extensive revision of that part of his book which describes the structure of Riemannian symmetric spaces, and the topics of differential geometry and Lie group theory necessary for its understanding. Chapter X of the earlier book, about ``Functions on symmetric spaces'', is omitted in the book under review; the author has announced that a second volume [Groups and geometric analysis, to appear] will be devoted to this subject, which has developed very intensively since 1962. For a basic description of the contents of the present book we refer to the review of the earlier version [op. cit., MR0145455]; we restrict ourselves here to describing the main complements and changes. Chapter I, ``Elementary differential geometry'', is completed by an appendix, containing some tools from general topology and a theorem about mappings of constant rank. The second chapter, ``Lie groups and Lie algebras'', now contains a section about invariant differential forms on Lie groups. Chapters III: ``Structures of semisimple Lie algebras'', IV: ``Symmetric spaces'', V: ``Decompositions of symmetric spaces'', and VI: ``Symmetric spaces of the noncompact type'', were only slightly changed, but Chapter VII: ``Symmetric spaces of the compact type'' was substantially completed: The Weyl group and its connection with geometric properties of the symmetric spaces are considered in detail, including the affine Weyl group, the theory of shortest geodesics and minimal totally geodesic spheres. In Chapter VIII: ``Hermitian symmetric spaces'', the bounded symmetric domains are now treated more completely. Chapter IX: ``Structure of semisimple Lie groups'', is newly included. It contains the Cartan, Iwasawa and Bruhat decompositions, the theory of real Cartan subalgebras and the theory of automorphisms of compact semisimple Lie algebras. These subjects and much other concrete material have been prepared with a view to applications to the classification of symmetric spaces and to harmonic analysis. The book concludes with Chapter X: ``The classification of simple Lie algebras and of symmetric spaces''. In contrast to the first edition, the classification is carried out with full proofs. The classification of simple Lie algebras over R is based on V. Kac's method for classifying automorphisms of finite order of simple Lie algebras over C [cf. Kac, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 94–96; MR0251091] which is, for the first time, described in detail in the book under review. As before, the book contains many exercises contributing interesting complements to the basic text. In the new edition, about 50 pages of ``Solutions to exercises'' are included. The book is excellently printed and contains apparatus useful for various groups of readers: suggestions for using the book as a textbook, along with a bibliography, list of symbols and notational conventions, and an extended index. The style of the book is precise and clear; the exposition is motivated by pointing out the main problems and by very interesting comments about the history of the subject. It will serve as a determining book for qualifying young mathematicians, and as a valuable and reliable basis for further research work in an important field of mathematics.
Reviewer: Sulanke, Rolf [form MathSciNet]