Vanishing Results for the Cotton Tensor on Gradient Quasi-Einstein Solitons

Abstract

In this paper we study on gradient quasi-Einstein solitons with a fourth-order vanishing condition on the Weyl tensor. More precisely, we show that for n ≥ 4, the Cotton tensor of any n-dimensional gradient quasi-Einstein soliton with fourth order f-divergence free Weyl tensor is flat, if the manifold is compact, or noncompact but the potential function satisfies some growth condition. As corollaries, some local characterization results for the quasi-Einstein metrics are derived.