Splitting and gluing constructions for geodesically equivalent pseudo-Riemannian metrics

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 $G_j^i:=g^{ik}{\bar{g}}_{kj}$ 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

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References in corpus (5)

• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• Complete Einstein metrics are geodesically rigid
• The principle of equivalence and projective structure in space-times
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• Metric connections in projective differential geometry

Cited by in corpus (2)

• Degree of mobility for metrics of lorentzian signature and parallel (0,2)-tensor fields on cone manifolds
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry