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**Differential geometry, Lie groups, and symmetric spaces.
Reprint with corrections of the 1978 original.**
*(English)*
Zbl 0993.53002

Graduate Studies in Mathematics. 34. Providence, RI: American Mathematical Society (AMS). xxvi, 641 p. (2001).

The present book is a reprinted and corrected version of the author’s original book [Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80 (1978; Zbl 0451.53038)] which is one of the most important and excellent textbooks and a reference work about contemporary differential geometry. It is devoted to describe the structure of Riemannian symmetric spaces and the topics of differential geometry and Lie group theory necessary for its understanding. In comparison with the previous version the author has corrected few inaccuracies and he has added a section that contains some footnotes, giving alternative and more elementary proofs of some result.

The book is strictly related to two other excellent volumes by the same author [Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs 83, American Mathematical Society (2000; Zbl 0965.43007) and Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs 39, American Mathematical Society (1994; Zbl 0809.53057)].

The book is strictly related to two other excellent volumes by the same author [Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs 83, American Mathematical Society (2000; Zbl 0965.43007) and Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs 39, American Mathematical Society (1994; Zbl 0809.53057)].

Reviewer: Anna Fino (Torino)

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

43-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis |

53C30 | Differential geometry of homogeneous manifolds |

53C35 | Differential geometry of symmetric spaces |

22E10 | General properties and structure of complex Lie groups |

22E15 | General properties and structure of real Lie groups |

22E20 | General properties and structure of other Lie groups |