Abstract
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is the simplest case, which was not covered by Levi-Civita’s theorem.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.
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Zelenko, I. On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1. J Math Sci 135, 3168–3194 (2006). https://doi.org/10.1007/s10958-006-0151-5
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DOI: https://doi.org/10.1007/s10958-006-0151-5