Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics

Bolsinov, Alexey; Matveev, Vladimir

doi:10.1090/S0002-9947-2011-05187-1

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics
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by Alexey V. Bolsinov and Vladimir S. Matveev PDF

Trans. Amer. Math. Soc. 363 (2011), 4081-4107 Request permission

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 generalise the Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics.

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