  We generalize the following classical result of Fubini for pseudo-Riemannian
metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same
unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly
nonproportional (i.e., the minimal polynomial of $g^{i\alpha} \bar g_{\alpha
j}$ coincides with the characteristic polynomial) at least at one point, then
they have constant curvature.
