Title: Hamiltonian systems of negative curvature are hyperbolic
Authors: A.A. Agrachev, N.N. Chtcherbakova
Journal title: Russian Math. Dokl.
The curvature and the reduced curvature are basic differential invariants of the pair: on the symplectic manifold. We show that negativity of the curvature implies that any bounded semi-trajectory of the Hamiltonian system tends to a hyperbolic equilibrium, while negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. Last statement generalizes a well-known property of the geodesic flows of Riemannian manifolds with negative sectional curvatures.