$m$-quasi Einstein manifolds with convex potential

TL;DR

This paper studies $m$-quasi Einstein manifolds with convex potential functions, proving conditions under which they have constant scalar curvature, are Einstein, and analyzing the properties of the potential function.

Contribution

It establishes new results on the scalar curvature and Einstein conditions of $m$-quasi Einstein manifolds with convex potentials, including integral and completeness conditions.

Findings

Manifolds with certain integral conditions have constant scalar curvature.

Potential function aligns with Hodge-de Rham potential up to a constant.

Complete non-compact manifolds with bounded scalar curvature have zero scalar curvature.

Abstract

The main objective of this paper is to investigate the $m$-quasi Einstein manifold when the potential function becomes convex. In this article, it is proved that an $m$-quasi Einstein manifold satisfying some integral conditions with vanishing Ricci curvature along the direction of potential vector field has constant scalar curvature and hence the manifold turns out to be an Einstein manifold. It is also shown that in an $m$-quasi Einstein manifold the potential function agrees with Hodge-de Rham potential up to a constant. Finally, it is proved that if a complete non-compact and non-expanding $m$-quasi Einstein manifold has bounded scalar curvature and the potential vector field has global finite norm, then the scalar curvature vanishes.