On curvature properties of certain quasi-Einstein hypersurfaces (Q2909468)

A manifold \((M,g)\) is called a quasi-Einstein manifold if its Ricci tensor \(S\) satisfies \(S=\alpha g+\epsilon w\otimes w\) at every point \(p\in M\), where \(w\) is a covector at \(p\), \(\alpha\in\mathbb R,\;\epsilon=\pm 1\). Considerations on quasi-Einstein hypersurfaces are closely related to the investigations on the Ryan problem of the equivalence on hypersurfaces in Euclidean spaces \(\mathbb E^{n+1}\), \(n\geqslant 4\), of the conditions of semisymmetry and Ricci-semisymmetry. As a generalization, one can consider on hypersurfaces in semi-Riemannian space-forms the problem of equivalence of the conditions of pseudosymmetry and Ricci-pseudosymmetry.NEWLINENEWLINEHere, the authors investigate quasi-Einstein hypersurfaces in semi-Riemannian space-forms satisfying some Walker-type identity. They prove that such hypersurfaces are Ricci-pseudosymmetric manifolds. Next, the authors consider pseudosymmetric hypersurfaces in space-forms and establish the equivalence of the quasi-Einstein condition to some curvature conditions. Basing on these results they construct an example of a quasi-Einstein non-pseudosymmetric and Ricci-pseudosymmetric warped product which locally can be realized as a hypersurface in a semi-Riemannian space-form.