A Conformal Quasi Einstein Characterization Of The Round Sphere

TL;DR

This paper characterizes the round sphere using conformal and Killing vector fields on quasi-Einstein manifolds, extending previous results and providing new integral identities for scalar curvature analysis.

Contribution

It extends Cochran's result to include the case m=-2, shows conformal vector fields are Killing on such manifolds, and derives a new integral identity for scalar curvature.

Findings

X is Killing if the integral of the Lie derivative of scalar curvature along X is non-positive

On closed manifolds, conformal vector fields are Killing, and non-Killing conformal fields imply the manifold is a sphere

A new integral identity for vector fields offers a direct proof of the Bourguignon-Ezin conservation identity

Abstract

We extend the following result of Cochran ``A closed $m$-quasi Einstein manifold ($M,g,X$) with $m \ne -2$ has constant scalar curvature if and only if $X$ is Killing" covering the missing accidental case $m=-2$ and generalize it showing that $X$ is Killing if the integral of the Lie derivative of the scalar curvature along $X$ is non-positive. For a closed $m$-quasi Einstein manifold of dimension $n \ge 2$, if $X$ is conformal, then it is Killing; and in addition, if $M$ admits a non-Killing conformal vector field $V$, then it is globally isometric to a sphere and $V$ is gradient for $n > 2$. Finally, we derive an integral identity for a vector field on a closed Riemannian manifold, which provides a direct proof of the Bourguignon-Ezin conservation identity.