  In this paper we prove that geodesic mappings of (pseudo-) Riemannian
manifolds preserve the class of differentiability \hbox{$(C^r, r\geq1)$}. 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. $\bar g={\rm const}\cdot g$.
