Abstract
In a complete Riemannian manifold
(M,g)
if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of
(M,g)
. In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds
(M,g)
admitting a real valued function
u
such that the hessian of
u
has atmost two eigenvalues
−u
and
−
u+1
2
, under some mild hypothesis on
(M,g)
. This generalises a well known result of Obata which characterizes all round spheres.