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.
In this paper, we study n-dimensional recurrent equiaffine projective Euclidean manifolds, i.e. manifolds with absolute recurrent curvature tensor, which admit geodesic mappings onto Euclidean space, and they are equiaffine (where was obtained the symmetric Ricci tensor). We obtained main conditions of recurrent projective Euclidean spaces and constructed their examples.
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