A Conformal Quasi Einstein Characterization Of The Round Sphere

TL;DR

This work extends Cochran's $m$-quasi Einstein framework to the singular case $m=-2$ and proves that, on closed manifolds, the integral condition $\int_M (\mathcal{L}_X R) \, dv_g \le 0$ forces $X$ to be Killing. It further shows that any conformal vector field on a closed $m$-quasi Einstein manifold is Killing, and that a non-Killing conformal field induces a global spherical geometry with the conformal field gradient up to Killing. The paper also provides a detailed analysis that yields a generalized Bourguignon-Ezin conservation identity for arbitrary vector fields and offers a direct derivation via divergence techniques, enhancing tools for conformal geometry on quasi-Einstein manifolds.

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.

Theorems & Definitions (12)

• Theorem 1.1
• Corollary 1.2
• Lemma 1.3
• proof
• Remark 1.4
• Remark 1.5
• Theorem 2.1
• proof
• Remark 2.2
• Theorem 2.3