We analyze how the transport by diffusion of a substance over a curved surface is affected by the local curvature. We show that these effects are completely characterized by the geodesic curvature kappa(g)(r) or equivalently by the Gaussian curvature K(r). The mean square geodesic distance traveled by a diffusing molecule on the surface and the probability distribution are obtained perturbately as a function of the surface curvature. The obtained corrections in the curvature are shown to be relevant in the understanding of lateral diffusion on biomembranes.