Locally conformally flat weakly-Einstein manifolds

Four-dimensional homogeneous manifolds satisfying some Einstein-like conditions

“…Locally conformally flat ρ-Einstein metrics which are not Einstein are either products M 1 (c)×M 2 (−c) with dim M 1 = dim M 2 , or otherwise they are locally isometric to warped products I × f R n−1 with metric dt 2 + f (t) 2 g 0 , where g 0 denotes the Euclidean metric on R n−1 and f (t) = 1) , where I ⊂ R is a real interval depending on the constants a, b ∈ R, a = 0. In contrast with the previous situations a locally conformally flat manifold is R[ρ]-Einstein if and only if it is the product manifold M 1 (c) × M 2 (−c) of two manifolds with constant opposite curvature and dim M 1 = dim M 2 [17]. Homogeneous Einstein metrics in dimension four were classified by Jensen [18], showing that they are either real space forms, complex space forms, or a product of two surfaces with the same constant Gauss curvature.…”

“…We emphasize the existence of non-Einsteinian weakly-Einstein metrics. For instance, Ř-Einstein locally conformally flat examples with non-constant scalar curvature have been constructed in [16]. Locally conformally flat ρ-Einstein metrics which are not Einstein are either products M 1 (c)×M 2 (−c) with dim M 1 = dim M 2 , or otherwise they are locally isometric to warped products I × f R n−1 with metric dt 2 + f (t) 2 g 0 , where g 0 denotes the Euclidean metric on R n−1 and f (t) = 1) , where I ⊂ R is a real interval depending on the constants a, b ∈ R, a = 0.…”