A Note On Obata's Rigidity Theorem I

2012/03/23 by Guoqiang Wu, Wu, Guoqiang, Rugang Ye +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG

paper · pdf · doi:10.48550/arxiv.1203.5307

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

openalex publication_date 2012/03/23 · arxiv created 2012/04/07 · arxiv updated 2012/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

Cited by

• Conformally weighted Einstein manifolds: the uniqueness problem

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