Geodesic mappings and some generalizations. (English) Zbl 1222.53002

Mikeš, Josef; Vanžurová, Alena; Hinterleitner, Irena

Olomouc: Palacký University, Faculty of Science (ISBN 978-80-244-2524-5/pbk). 304 p. (2009).

Authors’ abstract: “The book is devoted to the theory of geodesic mappings and their generalizations. The basics of the theory of conformal, concircular, geodesic, holomorphically-projective, F-planar and further mappings and transformations of spaces with affine connection, Riemannian and Kählerian spaces are presented in detail. The definitions, basic properties and classical results of the studied mappings and manifolds are given and supplemented by important new results in this areas, which were in the majority obtained by the authors.”
The book covers in particular the following topics: systems of partial differential equations of Cauchy type; Riemannian manifolds; Kähler manifolds; equidistant spaces; affine mappings and transformations; isometric mappings and transformations; homothetic mappings and transformations; conformal and isometric mappings; conformal mappings onto Einstein spaces; conformal transformations; geodesic mappings of manifolds with affine connection; geometric objects under geodesic mappings; geodesic mappings of equiaffine manifolds; projectively flat manifolds; geodesic mappings onto Riemannian manifolds; Mikeš-Berezovski equations; classical examples of geodesic mappings; geodesic mappings and equidistant spaces; geodesic mappings of spaces of constant curvature; geodesic mappings of Einstein spaces; geodesic mappings of pseudo-symmetric manifolds; generalized symmetric, recurrent and semisymmetric \({\mathbf V}_n\); geodesic mappings of Kähler manifolds; \(F\)-planar mappings of spaces with affine connections; \(F\)-planar mappings onto Riemannian manifolds; infinitesimal \(F\)-planar transformations; holomorphically projective mappings of Kähler manifolds.

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C22	Geodesics in global differential geometry
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53A30	Conformal differential geometry (MSC2010)
53B21	Methods of local Riemannian geometry

Keywords:

Riemannian manifolds; Kähler manifolds; equidistant spaces; affine mappings and transformations; isometric mappings and transformations; homothetic mappings and transformations; conformal and isometric mappings; conformal mappings onto Einstein spaces; conformal transformations; geodesic mappings; equiaffine manifolds; Mikeš-Berezovski equations; Einstein spaces; F-planar mappings; projective mappings; Kähler manifolds