  The general covariance of the diffusion equation is exploited in order to
explore the curvature effects appearing on brownian motion over a d-dimensional
curved manifold. We use the local frame defined by the so called Riemann normal
coordinates to derive a general formula for the mean-square geodesic distance
(MSD) at the short-time regime. This formula is written in terms of $O(d)$
invariants that depend on the Riemann curvature tensor. We study the
n-dimensional sphere case to validate these results. We also show that the
diffusion for positive constant curvature is slower than the diffusion in a
plane space, while the diffusion for negative constant curvature turns out to
be faster. Finally the two-dimensional case is emphasized, as it is relevant
for the single particle diffusion on biomembranes.
