Fubini Theorem for pseudo-Riemannian metrics

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 $g^{iα} \bar g_{αj}$ coincides with the characteristic polynomial) at least at one point, then they have constant curvature.

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