On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1 |
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Authors: | I Zelenko |
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Institution: | (1) S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy |
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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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Keywords: | |
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