Geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability

Abstract

In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox{$(C^r,r\ge 1)$}. Also, if the Einstein space $V_n$ admits a non trivial geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold ${\bar{V}}_n\in C^1$, then ${\bar{V}}_n$ is an Einstein space. If a four-dimensional Einstein space with non constant curvature globally admits a geodesic mapping onto a (pseudo-) Riemannian manifold ${\bar{V}}_4\in C^1$, then the mapping is affine and, moreover, if the scalar curvature is non vanishing, then the mapping is homothetic, i.e. $\bar{g}=\mathrm{const}\cdot g$.

8 pages

References in corpus (1)

• On geodesic mappings of manifolds with affine connection