In this paper, we consider and resolve a geometric problem by using μ（z）-homeomorphic theory, which is the generalization of quasiconformal mappings. A sufficient condition is given such that a Ct-two-real-dimensional connected orientable manifold with almost positive definite metric can be made into a Riemann surface by the method of isothermal coordinates. The result obtained here is actually a generalization of Chern＇s work in 1955.
