A Generalisation of Obata's theorem

Details A Generalisation of Obata's theorem A Generalisation of Obata's theorem

1995-11-23

arxiv.org/abs/dg-ga/9511010v1

Mathematics

Differential Geometry

20 pages, uses Latex

Scientific paper

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.

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