This volume provides an introduction to the geometry of manifolds
equipped with additional structures connected with the notion of
symmetry. The content is divided into five chapters. Chapter I
presents the elements of differential geometry which are used in
subsequent chapters. Part of the chapter is devoted to general
topology, part to the theory of smooth manifolds, and the remaining
sections deal with manifolds with additional structures. Chapter II
is devoted to the basic notions of the theory of spaces. One of the
main topics here is the realization of affinely connected symmetric
spaces as totally geodesic submanifolds of Lie groups. In Chapter
IV, the most important classes of vector bundles are constructed.
This is carried out in terms of differential forms. The geometry of
the Euler class is of special interest here. Chapter V presents
some applications of the geometrical concepts discussed. In
particular, an introduction to modern methods of integration of
nonlinear differential equations is given, as well as
considerations involving the theory of hydrodynamic-type Poisson
brackets with connections to interesting algebraic structures. For
mathematicians and mathematical physicists wishing to obtain a good
introduction to the geometry of manifolds. This volume can also be
recommended as a supplementary graduate text.
General
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