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


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.


Maximum Principle Elementary Proof Riemannian Metrics Pontryagin Maximum Principle Contact Distribution 
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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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