Geodesic flows are standard models of conservative (as opposed to dissipative) dynamics. Large classes of mechanical systems can be considered as geodesic flows. This point of view is most fruitful when the underlying riemannian metric has enough local symmetries or when it has negative sectional curvature. The latter case belongs to the realm of chaotic dynamics.
We study geodesic flows of Weyl connections (generalizations of Levi-Civita connections of the riemannian case). A linear symmetric connection on the tangent bundle is called Weyl if the parallel transport preserves angles between vectors. These flows are models of systems out of equilibrium. They were originally considered in physics literature under the name of gaussian thermostats. The systems have a forcing term accompanied by the thermostatting term based on the Gauss' Least Constraint Principle for nonholonomic constraints. We revealed the geometric nature of the systems with gausssian thermostats. We showed that also in this generality if the sectional curvatures are negative the system has some hyperbolic properties. However we do not know if it is enough to force the uniform hyperbolicity (Anosov property). To settle such questions we propose to study Weyl connections with (local) symmetries. This project has the potential of delivering new classes of systems with hyperbolic dynamics, and/or classifying homogeneous Weyl connections with nonpositive sectional curvatures.
