Local normal forms for geodesically equivalent pseudo-Riemannian metrics

Abstract

Two pseudo-Riemannian metrics $g$ and $\bar{g}$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.

this is an extended version of the submitted paper. It contains all proofs with all details and also an appendix where we explain how can one construct a complex structure by a (1,1)-tensor such that all its eigenvalues are not real and such that the Nijenhuis torsion vanishes

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