Complete Einstein metrics are geodesically rigid
  We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and $\bar g$ coincide. 
Volodymyr Kiosak, Vladimir S. Matveev