Brownian motion meets Riemann curvature

The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing in 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 single-particle diffusion on biomembranes.