$m$-quasi Einstein manifolds with subharmonic potential

Abstract

The main objective of this paper is to investigate the $m$-quasi Einstein manifold when the potential function becomes subharmonic. 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.