On the equivalence of Ricci-semisymmetry and semisymmetry
Introduction. A semi-Riemannian manifold (M, g), n = dim M ≥ 3, is said to be semisymmetric  if (1) R · R = 0 holds on M . It is well known that the class of semisymmetric manifolds includes the set of locally symmetric manifolds (∇R = 0) as a proper subset. Recently the theory of Riemannian semisymmetric manifolds has been presented in the monograph . It is clear that every semisymmetric manifold satisfies The semi-Riemannian manifold (M, g), n ≥ 3, satisfying (2) is called
... etric. There exist non-semisymmetric Ricci-semisymmetric manifolds. However, under some additional assumptions, (1) and (2) are equivalent for certain manifolds. For instance, we have the following statement. Remark 1.1. (1) and (2) are equivalent on every 3-dimensional semi-Riemannian manifold as well as at all points of any semi-Riemannian manifold (M, g), of dimension ≥ 4, at which the Weyl tensor C of (M, g) vanishes (see e.g. [15, Lemma 2]). In particular, (1) and (2) are equivalent for every conformally flat manifold. It is a long standing question whether (1) and (2) are equivalent for hypersurfaces of Euclidean spaces; cf. Problem P 808 of  by P. J. Ryan, and references therein. More generally, one can ask the same question for hypersurfaces of semi-Riemannian space forms. It was proved in  that 1991 Mathematics Subject Classification: 53B20, 53B30, 53B50, 53C25, 53C35, 53C80.