Splitting and gluing constructions for geodesically equivalent pseudo-Riemannian metrics

TL;DR

This paper introduces new constructions for analyzing geodesically equivalent pseudo-Riemannian metrics, simplifying classification problems by reducing them to cases with specific eigenvalue structures, and generalizes a key hierarchy theorem.

Contribution

It presents two novel constructions that reduce classification and Lie problem challenges to simpler eigenvalue cases, and extends the Topalov-Sinjukov hierarchy theorem to pseudo-Riemannian metrics.

Findings

Reduction of classification problems to eigenvalue cases

Generalization of the Topalov-Sinjukov hierarchy theorem

Initial applications demonstrating the effectiveness of the constructions

Abstract

Two metrics $g $ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the (1,1)-tensor $G^i_j:= g^{ik} \bar g_{kj}$ has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalize Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics