Bach-flat h-almost gradient Ricci solitons

TL;DR

This paper studies a generalization of gradient Ricci solitons called h-almost gradient Ricci solitons, proving that Bach-flatness and certain conditions imply the manifold is Einstein or rigid, with special properties in four dimensions.

Contribution

It establishes new rigidity results for Bach-flat h-almost gradient Ricci solitons, including conditions leading to Einstein or conformally flat structures.

Findings

Manifolds are Einstein or rigid under Bach-flatness and positivity of dh/du.

Such manifolds have harmonic Weyl curvature.

In four dimensions, the metric is conformally flat.

Abstract

On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat.