Differential geometry, Lie groups, and symmetric spaces
This book is intended as a textbook and reference work. It begins with a general self-contained exposition of differential and Riemannian geometry; affine connections, exponential mapping, geodesics, and curvature are discussed. Then the basic theory of Lie groups and Lie algebras, homogeneous spaces, the adjoint group, etc., are developed. The preliminary structure theory of semisimple groups is considered with emphasis on compact real forms. Next, an introductory geometric study of symmetric spaces is given. Then the local decomposition of a symmetric space into R/sup n/ and the two main types of symmetric spaces, the compact type and the noncompact type, is investigated. Symmetric spaces of noncompact type are studied in greater detail; since these spaces are completely determined by their isometry group, this chapter is primarily a global study of noncompact semisimple Lie groups. Next, topological and differential geometric properties of the compact symmetry space U/K are derived by study of the isotropy action of K on U/K and on its tangent space at the origin. Then Hermitian symmetric spaces are considered; the emphasis is on the noncompact ones and the Cartan-Harish-Chandra representation of these as bounded domains. After a more intense study of noncompact semisimple Lie groups, the book concludes with a classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over C and Cartan's classification of simple Lie algebras over R. Each chapter begins with a short summary and ends with references to source materials. 617 references, 2 figures, 10 tables. (RWR)
- OSTI ID:
- 6662393
- Country of Publication:
- United States
- Language:
- English
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