Classification of 4-dimensional homogeneous weakly einstein manifolds

A curvature identity on a~6-dimensional Riemannian manifold and its applications

“…Based on our current work, the definition 4-dimensional weakly Einstein manifold introduced in our paper [11,12] may be made more precise and the definition becomes 4-dimensional weakly Einstein manifold of degree 4. We note that Arias-Marco and Kowalski recently obtained a classification theorem for 4-dimensional homogeneous weakly Einstein manifolds [1].…”

Homogeneous generalized Einstein manifolds in dimension four

“…Proof. Let ⟨• , •⟩ be a left-invariant metric as described in (6). In order to simplify the notation, we set where the coefficients 𝔚 i𝑗 are polynomials on the structure constants given by…”

“…It was shown by Jensen 5 that any four‐dimensional homogeneous Einstein metric is locally symmetric and thus isometric to a (real or complex) space form or to the product of two real space forms with the same sectional curvature. Homogeneous four manifolds satisfying the weakly Einstein condition have been classified in Arias‐Marco and Kowalski, 6 showing that in the non‐Einstein case, they are homothetic to or to the left‐invariant metric on determined by the Lie algebra where { e 1 , e 2 , e 3 , e 4 } is an orthonormal basis. Locally conformally flat homogeneous manifolds are symmetric due to Takagi 7 and thus isometric to a real space form or a product , where N ( c ) is a three‐dimensional space form, or a product of two space forms of constant opposite sectional curvature.…”

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

“…Remark 1 in Section 2). We refer the readers to [1,15,16,17,18,19] for more results on this subject. Here, we shall replace the assumption of Einstein in the Miao Tam result (see Theorem 1) by the weakly Einstein condition, which is weaker that the former one in low dimension.…”

“…Remark 1 in Section 2). We refer the readers to [1,15,16,17,18,19] for more results on this subject.…”