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.
In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n + 1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with afine connection onto symmetric manifolds.
For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with the metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to continue investigations concerning existence of almost geodesic mappings of manifolds with linear (affine) connection, particularly of the so-calledπ 1 mappings, i.e. canonical almost geodesic mappings of type π 1 according to Sinyukov. First we give necessary and sufficient conditions for existence ofπ 1 mappings of a manifold endowed with a linear connection onto pseudo-Riemannian manifolds. The conditions take the form of a closed system of PDE's of first order of Cauchy type. Further we deduce necessary and sufficient conditions for existence ofπ 1 mappings onto generalized Ricci-symmetric spaces. Our results are generalizations of some previous theorems obtained by N.S. Sinyukov.
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