A Note On Obata's Rigidity Theorem I

TL;DR

This paper extends Obata's rigidity theorem to broader contexts, including generalized equations and different geometries, while analyzing the manifold's geometry and topology.

Contribution

It introduces new rigidity results for the generalized Obata equation and explores hyperbolic and Euclidean analogs, expanding the theorem's applicability.

Findings

Rigidity theorems for generalized Obata equations

Characterization of manifold geometry and topology

Extensions to hyperbolic and Euclidean settings

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.