A Note On Obata's Rigidity Theorem I

Details A Note On Obata's Rigidity Theorem I A Note On Obata's Rigidity Theorem I

2012-03-23

arxiv.org/abs/1203.5307v2

Mathematics

Differential Geometry

Some remarks are added. A few typos are removed. The proof of one lemma is modified

Scientific paper

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.

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