Abstract: In this paper , for surfaces with constant mean curvature, we studied their global propert ies. When they satisfy some pinching conditions, results are that :1) If M be a topological two sphere, then either M is flat , totally umbilic or M is a minimal surface. 2) If M be a totally real minimal surface of a ka..hler suface, then either M is the standard embedding of RP² into CP² or M is totally geodesic or flat.

Key words: surface, mean curvature, constant curved manifold, flat, totally umbilic, minimal surface, holomorphic sectional curvature