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

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 $(M^n,g)$, $(n\ge 4),$ 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., ${\mathrm{div}}^{4}W=0$).

12 pages

References in corpus (1)

• Rigidity of Gradient Shrinking Ricci Solitons