  Our work proposes a unified approach to three different topics in a general
Riemannian setting: splitting theorems, symmetry results and overdetermined
elliptic problems. By the existence of a stable solution to the semilinear
equation $-\Delta u = f(u)$ on a Riemannian manifold with non-negative Ricci
curvature, we are able to classify both the solution and the manifold. We also
discuss the classification of monotone (with respect to the direction of some
Killing vector field) solutions, in the spirit of a conjecture of De Giorgi,
and the rigidity features for overdetermined elliptic problems on submanifolds
with boundary.
