Quasi-Bach flow and quasi-Bach solitons on Riemannian manifolds

Abstract

In this article, we introduce quasi-Bach tensor and correspondingly introduce almost quasi-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We explore some properties of gradient quasi Bach solitons with harmonic Weyl curvature tensor. We study the relationships between Weyl tensor, Cotton tensor, tensor $D$ introduced by Cao, and quasi Bach tensor. We also find the evolution of volume, Einstein metric, Ricci curvature and scalar curvature, under the quasi Bach flow. Our results obtained here extends the results of Bach solitons and Bach flow. Finally, we obtain the characterization of gradient quasi-Bach soliton of type I, a particular quasi Bach soliton, on the product manifolds $S^2\times H^2$ and $R^2\times H^2$. Our exploration generalizes gradient Bach soliton on $R^2\times H^2$ obtained by P. T. Ho, while the gradient soliton on $S^2\times H^2$ is a novel one and is complementary to the results obtained by P. T. Ho.