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Introduction to the geometry of manifolds with symmetries. (Vvedenie v geometriyu mnogoobrazij s simmetriyami.) (Russian) Zbl 0705.53002

Moskva: Izdatel’stvo Moskovskogo Gosudarstvennogo Universiteta. 360 p. R. 5.00 (1989).
The book grew out of lectures given by the author at Moscow University. Topics included: introduction to differential geometry, i.e. elements of general topology, manifolds, affine and Riemannian connections; Lie groups and Lie algebras over closed fields, symmetric spaces and their realization as a totally geodesic submanifold of a Lie group; the differential-geometric construction of characteristic classes of vector bundles; applications to nonlinear differential equations: Poisson brackets of hydrodynamical type and differential equations on Lie groups [see S. Novikov’s papers from 1980 to 1986 and also A. Fomenko, Symplectic geometry (Russian) (Izd. Mosk. Univ.); English translation (Gordon and Breach) (1988)].
The proofs are fully worked out, except in the last part; several topics are described in an original way and many interesting examples (but no exercises) are included. The text is accessible for students and it will be of interest to a wide audience of mathematicians and, especially, theoretical physicists.
Reviewer: M.Malakhal’tsev

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
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