  In this note we present various extensions of Obata's rigidity theorem
concerning the Hessian of a function on a Riemannian manifold. They include
general rigidity theorems for the generalized Obata equation, and hyperbolic
and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity
case we also characterize the geometry and topology of the underlying manifold
in more general situations.
