On the classification of noncompact steady quasi-Einstein manifold with vanishing condition on the Weyl tensor

TL;DR

This paper classifies certain noncompact steady quasi-Einstein manifolds with specific Weyl tensor conditions, showing they are warped products with Einstein fibers under nonnegative Ricci curvature.

Contribution

It proves that steady m-quasi-Einstein manifolds with fourth-order divergence-free Weyl tensor are warped products with Einstein fibers, extending classification results under new curvature conditions.

Findings

Manifolds are warped products with Einstein fibers.

Conditions on the Weyl tensor lead to classification.

Results apply to simply connected manifolds with nonnegative Ricci curvature.

Abstract

The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with $(n-1)-$dimensional Einstein fiber, provided that $M$ has fourth order divergence-free Weyl tensor (i.e., ${\rm div}^{4}W=0$).