A Generalisation of Obata's theorem

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 $-{{u+1}\over 2}$, under some mild hypothesis on $(M, g)$. This generalises a well known result of Obata which characterizes all round spheres.