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

On projectively equivalent metrics near points of bifurcation