  This paper aims at investigating the influence of space-time curvature on the
uncertainty relation. In particular, relying on previous findings, we assume
the quantum wave function to be confined to a geodesic ball on a given
space-like hypersurface whose radius is a measure of the position uncertainty.
On the other hand, we concurrently work out a viable physical definition of the
momentum operator and its standard deviation in the non-relativistic limit of
the 3+1 formalism. Finally, we evaluate the uncertainty relation which to
second order depends on the Ricci scalar of the effective 3-metric and the
corresponding covariant derivative of the shift vector. For the sake of
illustration, we apply our general result to a number of examples arising in
the context of both general relativity and extended theories of gravity.
