  The appearance of a geometric flow in the conservation law of particle number
in classical particle diffusion and in the conservation law of probability in
quantum mechanics is discussed in the geometrical environment of a
two-dimensional curved surface with thickness $\epsilon$ embedded in $R_3$.
  In such a system with a small thickness $\epsilon$, the usual two-dimensional
conservation law does not hold and we find an anomaly. The anomalous term is
obtained by the expansion of $\epsilon$. We find that this term has a Gaussian
and mean curvature dependence and can be written as the total divergence of
some geometric flow. We then have a new conservation law by adding the
geometric flow to the original one. This fact holds in both classical diffusion
and quantum mechanics when we confine particles to a curved surface with a
small thickness.
