Fubini Theorem for pseudo-Riemannian metrics

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.

References in corpus (3)

• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• Invariant characterization of Liouville metrics and polynomial integrals
• Metrisability of three-dimensional projective structures

Cited by in corpus (3)

• Complete Einstein metrics are geodesically rigid
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Riemannian metrics having common geodesics with Berwald metrics