A Note on Obata’s Rigidity Theorem

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 rigidit...

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Veröffentlicht in:Communications in mathematics and statistics 2014-12, Vol.2 (3-4), p.231-252
Hauptverfasser: Wu, Guoqiang, Ye, Rugang
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:2194-6701
2194-671X
DOI:10.1007/s40304-014-0038-6