Let $S$ be a fixed compact oriented surface with negative Euler characteristic (i.e. of genus at least 2). By The Uniformization Theorem, the only interesting Riemannian structures are hyperbolic metrics (i.e. Riemannian metrics with constant Gaussian curvature $-1$). Teichmuller space Teich(S) of $S$ is the isotopy classes of hyperbolic metrics on the surface $S.$ In this talk, we will first explain $Teich(S)$ by using the classical Fenchel-Nielsen coordinates. Then, we will explain the \emp{shearing coordinates} of $Teich(S),$ introduced by W. Thurston and F. Bonahon. These coordinates are associated with the choice of a maximal geodesic lamination $\lambda $ on the surface $S.$ Geodesic lamination $\lambda \subset S$ is a closed subset of the surface $S$ which can be decomposed as a union of disjoint geodesics with no self-intersection. If time permits, we would like to state some facts on $Teich(S),$ where we used the shearing coordinates.

Yasar Sozen

General Seminar, 24.11.2005