On the classification of 4-dimensional $(m,\rho)$-quasi-Einstein manifolds with harmonic Weyl curvature

TL;DR

This paper classifies 4-dimensional $(m, ho)$-quasi-Einstein manifolds with harmonic Weyl curvature, revealing their local geometric structures and providing a broader understanding of Einstein-type manifolds.

Contribution

It offers a new local classification of $(m, ho)$-quasi-Einstein manifolds with harmonic Weyl curvature, including complete cases and special subclasses.

Findings

Classified local structures as products of constant curvature surfaces

Identified conditions for singular and conformally flat metrics

Extended results to gradient Einstein-type manifolds

Abstract

In this paper we study 4-dimensional $(m,\rho)$-quasi-Einstein manifolds with harmonic Weyl curvature when $m\notin\{0,\pm1,-2,\pm\infty\}$ and $\rho\notin\{\frac{1}{4},\frac{1}{6}\}$. We prove that a non-trivial $(m,\rho)$-quasi-Einstein metric $g$ (not necessarily complete) is locally isometric to one of the followings: (i) $\mathcal{B}^2_\frac{R}{2(m+2)}\times \mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$ where $\mathcal{B}^2_\frac{R}{2(m+2)}$ is a northern hemisphere in the 2-dimensional sphere $\mathbb{S}^2_\frac{R}{2(m+2)}$, $\mathbb{N}_\delta$ is the 2-dimensional Riemannian manifold with constant curvature $\delta$ and $R$ is the constant scalar curvature of $g$, (ii) $\mathcal{D}^2_\frac{R}{2(m+2)}\times\mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$ where $\mathcal{D}^2_\frac{R}{2(m+2)}$ is one half (cut by a hyperbolic line) of the hyperbolic plane $\mathbb{H}^2_\frac{R}{2(m+2)}$, (iii) $\mathbb{H}^2_\frac{R}{2(m+2)}\times\mathbb{N}^2_\frac{R(m+1)}{2(m+2)}$, (iv) a certain singular metric with $\rho=0$, (vi) a locally conformally flat metric. By applying this local classification, we obtain a classification of complete $(m,\rho)$-quasi-Einstein manifolds under the harmonic Weyl curvature condition. Our result can be viewed as a local classification of gradient Einstein-type manifolds. One corollary of our result is the classification of $(\lambda,4+m)$-Einstein manifolds which can be viewed as $(m,0)$-quasi-Einstein manifolds.