  Based upon the Smoluchowski equation on curved manifolds three physical
observables are considered for the Brownian displacement, namely, geodesic
displacement, $s$, Euclidean displacement, $\delta{\bf R}$, and projected
displacement $\delta{\bf R}_{\perp}$. The Weingarten-Gauss equations are used
to calculate the mean-square Euclidean displacements in the short-time regime.
Our findings show that from an extrinsic point of view the geometry of the
space affects the Brownian motion in such a way that the particle's diffusion
is decelerated, contrasting with the intrinsic point of view where dynamics is
controlled by the sign of the Gaussian curvature [J. Stat. Mech. P08006
(2010)]. Furthermore, it is possible to give exact formulae for
$\left<\delta{\bf R}\right>$ and $\left<\delta{\bf R}^{2}\right>$ on spheres
and minimal surfaces, which are valid for all values of time. In the latter
case, surprisingly, Brownian motion corresponds to the usual diffusion in flat
geometries, albeit minimal surfaces have non-zero Gaussian curvature. Finally,
the two-dimensional case is emphasized due to its close relation to surface
self-diffusion in fluid membranes.
