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Abstract. In this paper we study fundamental properties of geodesic mappings with respect to the smoothness class of metrics. We show that geodesic mappings preserve the smoothness class of metrics. We study geodesic mappings of Einstein spaces.Mathematics Subject Classification (2010). 53C21; 53C25; 53B21; 53B30.

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.