  In this paper we study a nonlocal diffusion problem on a manifold. These kind
of equations can model diffusions when there are long range effects and have
been widely studied in Euclidean space.
  We first prove existence and uniqueness of solutions and a comparison
principle. Then, for a convenient rescaling we prove that the operator under
consideration converges to a multiple of the usual Heat-Beltrami operator on
the manifold. Next, we look at the long time behavior on compact manifolds by
studying the spectral properties of the operator. Finally, for the model case
of hyperbolic space we study the long time asymptotics and find a different and
interesting behavior.
