On the Theory of Geodesic Mappings of Einstein Spaces and their Generalizations

Abstract

In this paper we consider results of the theory of geodesic mappings of Einstein spaces and their generalizations.In 1925 H. Brinkmann found the metric of equidistant spaces and obtained conditions, when these spaces are Einstein spaces, resp. spaces of constant curvature. We introduce the conditions on these spaces when they are semisymmetric, pseudosymmetric, Ricci semisymmetric, Ricci pseudosymmetric and spaces Vn(B).A diffeomorphism f between Riemannian spaces Vn and Vn is called a geodesic mapping, if any geodesic line in Vn is mapped into a geodesic line in Vn. In 1954 N.S. Sinyukov proved that equidistant spaces admit geodesic mappings. Our constructions of a geodesic mapping of Einstein spaces with the Brinkmann metric proves that Petrov’s conjecture is not true.We formulate results by E. Beltrami, R. Couty, V.I. Golikov, S. Formella, V.A. Kiosak, T. Levi‐Civita, J. Mikes, A.Z. Petrov and A.V. Pogorelov about geodesic mappings of Einstein spaces and spaces of constant curvature.Further we introduce results on geodesic mappings for Riemannian spaces, which are generalized Einstein spaces and spaces of constant curvature. For instance symmetric, recurrent, generalized recurrent, semisymmetric, pseudosymmetric, Ricci semisymmetric, Ricci pseudosymmetric spaces, spaces with harmonic curvature, etc. These results were obtained by many authors: R. Deszcz, V.A. Kiosak, J. Mikes, N.S. Sinykov, E.N. Sinyukova, V.S. Sobchuk, etc.