Sub-Riemannian problem on the three-dimensional solvable Lie group SOLV + with right-invariant distribution

Abstract

In this article we consider sub-Riemannian problem on the three dimensional solvable Lie group SOLV + with right-invariant distribution. We constructed the Hamiltonian structure for the geodesic flow of Carnot-Caratheodory metrics via the Pontryagin maximum principle. Recently, a very relevant research problems geodesic flows on sub-Riemannian manifolds (see, for example, [5, 6]). Detailed theoretical aspects are reflected in [1]. In work [3] A. Agrachev and D. Barilari made classification of left-invariant structures on three-dimensional Lie groups. According to this classification, there are invariants of the sub-Riemannian geometry, implemented in four nonnilpotent solvable Lie groups: SOLV