  We first describe the action of the fundamental group of a closed surface of
variable negative curvature on the oriented geodesics in its universal covering
in terms of a naturally-defined flat connection whose holonomy lies in the
group of Hamiltonian diffeomorphisms of S^1 x R. Consideration of the holonomy
necessitates an extension from Riemannian to Finsler metrics. The second part
of the paper follows the Higgs bundle approach to flat connections adapted to
this infinite dimensional group and focuses on a family of metrics, relying on
a construction of O.Biquard, which is parametrized by the infinite-dimensional
space of CR functions on the unit circle bundle of a hyperbolic surface. This
generates an alternative approach to defining a connection and offers the
possibility of this vector space representing a moduli space which generalizes
and includes the classical Teichmueller space.
