Fubini Theorem for pseudo-Riemannian metrics

TL;DR

This paper extends Fubini's classical theorem to pseudo-Riemannian metrics, showing that if three metrics share unparametrized geodesics and two are strictly nonproportional at a point, then they have constant curvature.

Contribution

It generalizes Fubini's theorem to pseudo-Riemannian metrics, establishing conditions under which shared geodesics imply constant curvature.

Findings

Three metrics sharing unparametrized geodesics

Two strictly nonproportional metrics imply constant curvature

Generalization of classical Fubini theorem

Abstract

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.