On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

The derivation-commutator $R \cdot C - C \cdot R$ of a semi-Riemannian manifold $(M,g)$, $\dim M \geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.