  Brownian motion of free particles on curved surfaces is studied by means of
the Langevin equation written in Riemann normal coordinates. In the diffusive
regime we find the same physical behavior as the one described by the diffusion
equation on curved manifolds [J. Stat. Mech. (2010) P08006]. Therefore, we use
the latter in order to analytically investigate the whole diffusive dynamics in
compact geometries, namely, the circle and the sphere. Our findings are
corroborated by means of Brownian dynamics computer simulations based on a
heuristic adaptation of the Ermak-McCammon algorithm to the Langevin equation
along the curves, as well as on the standard algorithm, but for particles
subjected to an external harmonic potential, deep and narrow, that possesses a
"Mexican hat" shape, whose minima define the desired surface. The short-time
diffusive dynamics is found to occur on the tangential plane. Besides, at long
times and compact geometries, the mean-square displacement moves towards a
saturation value given only by the geometrical properties of the surface.
