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A note on hyperbolic flows in sub-Riemannian Structures

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be hyperbolic and show new examples of Anosov flows.