Abstract: The properties of moduli space and Teichmuller space has been studied mathematically and physically to show their importance and applications in metric structure. Remann considered the space M of all complex structures on an orientable surface modulo the action of orientations preserving diffeomorphisms and derived the dimension of these space which expressed as dimg. M=6g-6..... (1) Where g>equal to 2 is the genus of the topological surface. Moduli spaces of Riemann surfaces have also been studied in algebraic geometry by F.P. Gardina. The geometric invariant theory developed by Mumford is major achievement. Deligne and Mumford studied the projective property of the Moduli space and they showed that the moduli space is quasi-projective and can be compactified naturally by adding in the stable nodal surfaces. The classical matrics on the Teichmuller space and the moduli spaces have also been studied independently. Each metric has played important role in the study of the geometry and topology of the moduli and Teichmuller spaces. We consider some of them such as (i) Finsler metrics: (ii) The Teichmuller metric|| ||i (iii) the Kobayashi (iv) metric || ||k and the Carath eodory metric || ||c They are all complete metrics on the Teichmuller space and are invariant under the moduli group action.
Keywords: Teichmuller space, Moduli space, Fisher metrics, Curvature Poincare matric, and Kahler-Einstein metric