A Generalisation of Obata's theorem

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

20 pages, uses Latex

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