On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

Details On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1 On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

2004-06-07

arxiv.org/abs/math/0406111v1

Mathematics

Differential Geometry

30 pages, to appear in "Journal of Mathematical Sciences"

Scientific paper

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 Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an 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 classical Levi-Civita's Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics. Secondly 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 make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita's Theorem.

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