Quasi-Einstein manifolds with Harmonic Weyl curvature

TL;DR

This paper classifies higher-dimensional quasi-Einstein manifolds with harmonic Weyl curvature, extending previous four-dimensional results and providing new examples that are neither locally conformally flat nor D-flat.

Contribution

It extends classification results of quasi-Einstein manifolds with harmonic Weyl curvature to dimensions five and above, and introduces new non-trivially curved examples.

Findings

Classification of $n$-dimensional quasi-Einstein manifolds with harmonic Weyl curvature for $n \ugeq 5$

Construction of new examples not conformally flat or D-flat

Extension of previous four-dimensional classification results

Abstract

In this paper, we classify $n$-dimensional ($n\geq 5$) quasi-Einstein manifolds with harmonic Weyl curvature, thus extending the work of Shin \cite{Shin} in dimension four for quasi-Einstein manifolds and refining the work of He-Petersen-Wylie \cite{HPW}. As a consequence, we provide new examples of quasi-Einstein manifolds which are neither locally conformally flat nor D-flat in the sense of \cite{CC12}.