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

In this paper we study four-dimensional ((m,rho ))-quasi-Einstein manifolds with harmonic Weyl curvature when (mnotin {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.