A Teichmüller space for negatively curved surfaces

We describe the action of fundamental group a closed Finsler surface negative curvature on geodesics in universal covering terms flat symplectic connection and consider first order deformation theory about hyperbolic metric. A construction O.Biquard yields family metrics which give nontrivial deformations holonomy, extending representation from SL(2,R) into Hamiltonian diffeomorphisms S^1 x R, producing an infinite-dimensional version Teichmuller space contains classical one.