We examine the Voronoï diagram generated by a homogeneous Poisson point process on a Riemannian surface. More precisely, we show a link between the mean characteristics of a cell and the Gaussian curvature of the surface. We focus on the mean number of vertices and start by recalling the exact formula in the case of the sphere. We then generalize it to an arbitrary surface by giving a high intensity asymptotic expansion. The proof relies notably on classical comparison results from Riemannian geometry.