Using an empirical Tersoff-Brenner many-body potential for the carbon atoms and a model Lennard-Jones interaction between the adatom and carbon atoms, the diffusion of an adatom over curved surfaces of C nanotubes is studied by calculating the potential energy surface and performing molecular dynamics simulation. The average curvature of the surface is found to have an important influence on the diffusion of the adatom. Positive curvature increases the diffusion barrier and corrugates the potential energy surface, while the negative curvature smoothes the potential energy surface, therefore it lowers the diffusion barrier. We also find that nanotube helicity can play an important role on the diffusion path, thus the adatom has different diffusion path for the armchair and zig-zag nanotube. The nature of the curvature effect on the surface diffusion is connected with the strain effect.