Geodesic Mappings Onto Riemannian Manifolds and Differentiability

Abstract

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto (pseudo-)Riemannian manifolds. We proved that if a manifold with affine (or projective) connection of differentiability class $C^r (r\geq2)$ admits a geodesic mapping onto a (pseudo-)Riemannian manifold of class $C^1$, then this manifold belongs to the differentiability class $C^{r+1}$. From this result follows if an Einstein spaces admits non-trivial geodesic mappings onto (pseudo-)Riemannian manifolds of class $C^1$ then this manifold is an Einstein space, and there exists a common coordinate system in which the components of the metric of these Einstein manifolds are real analytic functions.