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

Abstract

In this paper we study 4-dimensional $(m,\rho )$-quasi-Einstein manifolds with harmonic Weyl curvature when $m\notin \{0,\pm 1,-2,\pm \infty \}$ 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) ${\mathcal{B}}_{\frac{R}{2(m+2)}}^2\times {\mathbb{N}}_{\frac{R(m+1)}{2(m+2)}}^2$ where ${\mathcal{B}}_{\frac{R}{2(m+2)}}^2$ is a northern hemisphere in the 2-dimensional sphere ${\mathbb{S}}_{\frac{R}{2(m+2)}}^2$, ${\mathbb{N}}_{\delta }$ is the 2-dimensional Riemannian manifold with constant curvature $\delta$ and $R$ is the constant scalar curvature of $g$, (ii) ${\mathcal{D}}_{\frac{R}{2(m+2)}}^2\times {\mathbb{N}}_{\frac{R(m+1)}{2(m+2)}}^2$ where ${\mathcal{D}}_{\frac{R}{2(m+2)}}^2$ is one half (cut by a hyperbolic line) of the hyperbolic plane ${\mathbb{H}}_{\frac{R}{2(m+2)}}^2$, (iii) ${\mathbb{H}}_{\frac{R}{2(m+2)}}^2\times {\mathbb{N}}_{\frac{R(m+1)}{2(m+2)}}^2$, (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.

arXiv admin note: text overlap with arXiv:1604.02827 by other authors