  We present a geometric formula of Poincar\'e type, which is inspired by a
classical work of Sternberg and Zumbrun, and we provide a classification result
of stable solutions of linear elliptic problems with nonlinear Robin conditions
on Riemannian manifolds with nonnegative Ricci curvature.
  The result obtained here is a refinement of a result recently established by
Bandle, Mastrolia, Monticelli and Punzo.
