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

Classification of stable solutions for boundary value problems with  nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci  curvature