Diffeomorphism of affine connected spaces which preserved Riemannian and Ricci curvature tensors

Abstract

In this paper we study the preserving of Riemannian and Ricci tensors with respect to a diffeomorphism of spaces with affine connection. We consider geodesic and almost geodesic mappings of the first type. The basic equations of these maps form a closed system of Cauchy type in covariant derivatives. We determine the quantity of essential (substantial) parameters on which the general solution of this problem depends.

Basic invariants of geometric mappings

“…Some of them are the Thomas projective parameter, the Weyl projective tensor, the Weyl conformal curvature, and many others. These invariants may be found in the next monographs, books and papers: Mikeš [1,2,[8][9][10][11], Sinyukov [16], Hinterleitner [10], Berezovski [1,2,10], etc.…”

“…This invariant is important for researches about invariants of mappings characterized by deformation tensors P i jk which are not expressed in the form (2.3). The almost geodesic mappings of the first kind are an example of maps such that see [1,2,10] .…”

“…From the above obtained results, we will search invariants of equitorsion geodesic mappings defined on a generalized Riemannian space GR N . Furthermore, we will obtain an associated basic invariant of Weyl type of an almost geodesic mapping [1,2,10] defined on a Riemannian space R N . These invariants will be applied in the examples after theoretical researches.…”

“…Many authors have continued the Thomas's and Weyl's works. J. Mikeš [1,2,[8][9][10][11], I. Hinterleitner [10,11], N. S. Sinyukov [16], are some of them. Some of invariant geometrical object for diffeomorphisms of non-symmetric affine connection spaces are obtained in [15,18,19,21].…”

Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

“…For geodesic mappings of generalized symmetric and recurrent spaces, such problems were solved by J. Mikeš, V.S. Sobchuk, and others [21][22][23][24][25][26][27][38][39][40][41][42][43][44][45][46][47][48]. There are many works devoted to issues of the theory of geodesic mappings, for example [49][50][51][52][53][54][55][56][57][58].…”