%0 Journal Article
%J J. Math. Sci. 135 (2006) 3168-3194
%D 2006
%T On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1
%A Igor Zelenko
%X 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.
%B J. Math. Sci. 135 (2006) 3168-3194
%G en_US
%U http://hdl.handle.net/1963/2205
%1 2039
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-11T13:59:27Z\\nNo. of bitstreams: 1\\n0406111v1.pdf: 375215 bytes, checksum: cc4213e75a28857f98f01a294abff5e0 (MD5)
%R 10.1007/s10958-006-0151-5