In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay-Percus' projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic two-dimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley's coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetric channel, with modifications according to the metric terms. This expression is given as D(ξ)=D(0)/w'(ξ)√(g(1)/g(2)){arctan[√(g(2)/g(1))(y(0)'(ξ)+w'(ξ)/2)]-arctan[√(g(2)/g(1))(y(0)'(ξ)-w'(ξ)/2)]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces.