Archive | 2019
Differential Geometry of Special Mappings
Abstract
The monograph deals with the theory of conformal, geodesic,\nholomorphically projective, F-planar and others mappings and\ntransformations of manifolds with affine connection,\nRiemannian, Kahler and Riemann-Finsler manifolds. Concretely,\nthe monograph treats the following: basic concepts of\ntopological spaces, the theory of manifolds with affine\nconnection (particularly, the problem of semigeodesic\ncoordinates), Riemannian and Kahler manifolds (reconstruction\nof a metric, equidistant spaces, variational problems in\nRiemannian spaces, SO(3)-structure as a model of statistical\nmanifolds, decomposition of tensors), the theory of\ndifferentiable mappings and transformations of manifolds (the\nproblem of metrization of affine connection, harmonic\ndiffeomorphisms), conformal mappings and transformations\n(especially conformal mappings onto Einstein spaces, conformal\ntransformations of Riemannian manifolds), geodesic mappings\n(GM; especially geodesic equivalence of a manifold with affine\nconnection to an equiaffine manifold), GM onto Riemannian\nmanifolds, GM between Riemannian manifolds (GM of equidistant\nspaces, GM of Vn(B) spaces, its field of symmetric linear\nendomorphisms), GM of special spaces, particularly Einstein,\nKahler, pseudosymmetric manifolds and their generalizations,\nglobal geodesic mappings and deformations, GM between\nRiemannian manifolds of different dimensions, global GM,\ngeodesic deformations of hypersurfaces in Riemannian spaces,\nsome applications of GM to general relativity, namely three\ninvariant classes of the Einstein equations and geodesic\nmappings, F-planar mappings of spaces with affine connection,\nholomorphically projective mappings (HPM) of Kahler manifolds\n(fundamental equations of HPM, HPM of special Kahler manifolds,\nHPM of parabolic Kahler manifolds, almost geodesic mappings,\nwhich generalize geodesic mappings, Riemann-Finsler spaces and\ntheir geodesic mappings, geodesic mappings of Berwald spaces\nonto Riemannian spaces.