Conformally geodesic mappings satisfying a certain initial condition

summary:In this paper we study conformally geodesic mappings between pseudo-Riemannian manifolds $(M, g)$ and $(\bar{M}, \bar{g})$, i.e. mappings $f\colon M \rightarrow \bar{M}$ satisfying $f = f_1 \circ f_2 \circ f_3$, where $f_1, f_3$ are conformal mappings and $f_2$ is a geodesic mapping. Suppose that the initial condition $f^* \bar{g} = k g$ is satisfied at a point $x_0 \in M$ and that at this point the conformal Weyl tensor does not vanish. We prove that then $f$ is necessarily conformal