01442nas a2200097   4500008004300000245010600043210006900149520107200218100001801290856003601308       2006                        en_Ud  00aOn geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 10 ageodesic equivalence of Riemannian metrics and subRiemannian met3 aThe 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.1 aZelenko, Igor  uhttp://hdl.handle.net/1963/220501353nas a2200121   4500008004300000245003200043210003200075260001300107520103200120100002501152700001801177856003601195       2002                        en_Ud  00aGeometry of Jacobi Curves I0 aGeometry of Jacobi Curves I  bSpringer3 aJacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".1 aAgrachev, Andrei, A.1 aZelenko, Igor  uhttp://hdl.handle.net/1963/311000314nas a2200109   4500008004100000245003300041210003300074260001800107100002500125700001800150856003600168       2002                        en  d00aGeometry of Jacobi curves II0 aGeometry of Jacobi curves II  bSISSA Library1 aAgrachev, Andrei, A.1 aZelenko, Igor  uhttp://hdl.handle.net/1963/1589