  In this paper we provide some local and global splitting results on complete
Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting
through the analysis of some pointwise inequalities of Modica type which hold
true for every bounded solution to a semilinear Poisson equation. More
precisely, we prove that the existence of a nonconstant bounded solution $u$
for which one of the previous inequalities becomes an equality at some point
leads to the splitting results as well as to a classification of such a
solution $u$.
