  This paper is devoted to the study of rigidity properties for special
solutions of nonlinear elliptic partial differential equations on smooth,
boundaryless Riemannian manifolds. As far as stable solutions are concerned, we
derive a new weighted Poincar\'e inequality which allows to prove Liouville
type results and the flatness of the level sets of the solution in dimension 2,
under suitable geometric assumptions on the ambient manifold.
