  We investigate existence and nonexistence of stationary stable nonconstant
solutions, i.e. patterns, of semilinear parabolic problems in bounded domains
of Riemannian manifolds satisfying Robin boundary conditions. These problems
arise in several models in applications, in particular in Mathematical Biology.
We point out the role both of the nonlinearity and of geometric objects such as
the Ricci curvature of the manifold, the second fundamental form of the
boundary of the domain and its mean curvature. Special attention is devoted to
surfaces of revolution and to spherically symmetric manifolds, where we prove
refined results.
