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Journal of Mathematical Sciences

, Volume 135, Issue 4, pp 3168–3194 | Cite as

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

  • I. Zelenko
Article

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.

Keywords

Maximum Principle Elementary Proof Riemannian Metrics Pontryagin Maximum Principle Contact Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • I. Zelenko
    • 1
  1. 1.S.I.S.S.A.TriesteItaly

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