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The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3,formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformalcurvature tensor C, under some assumptions, can be expressed as a linearcombination 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 afamily of generalized Einstein metric conditions. In this survey paper wepresent recent results on manifolds and submanifolds, and in particularhypersurfaces, satisfying such conditions.

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions