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

Abstract

In this paper we study four-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 following: (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 the northern hemisphere in the two-dimensional (2D) sphere ${\mathbb {S}}^2_\frac{R}{2(m+2)}$, ${\mathbb {N}}_\delta$ is a 2D 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 half (cut by a hyperbolic line) of 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 $\rho =0$. (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete $(m,\rho )$-quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of $(\lambda ,4+m)$-Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.