A note on hyperbolic flows in sub-Riemannian Structures

TL;DR

This paper investigates the relationship between curvature invariants and hyperbolic behavior in Hamiltonian flows within sub-Riemannian structures, providing conditions for hyperbolicity and new examples of Anosov flows.

Contribution

It establishes a link between reduced curvature negativity and hyperbolicity, and introduces new conditions and examples of hyperbolic flows in sub-Riemannian geometry.

Findings

Negativity of reduced curvature implies hyperbolicity of invariant sets.

Provides sufficient conditions for hyperbolic reduced flows.

Introduces new examples of Anosov flows in sub-Riemannian contexts.

Abstract

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.