  Let Riemannian metrics $g$ and $\bar g$ on a connected manifold
  $M^n$ have the same geodesics (considered as unparameterized curves). Suppose
the eigenvalues of one metric with respect to the other are all different at a
point. Then, by the famous Levi-Civita's Theorem, the metrics have a certain
standard form near the point. Our main result is a generalization of
Levi-Civita's Theorem for the points where the eigenvalues of one metric with
respect to the other bifurcate.
