On some generalized Einstein metric conditions on hypersurfaces in semi-Riemannian space forms

Abstract

Abstract. Solutions of the P. J. Ryan problem as well as investigations of curvature properties of Cartan hypersurfaces and Ricci-pseudosymmetric hypersurfaces lead to curvature identities holding on every hypersurface M isometrically immersed in a semiRiemannian space form. These identities, under some assumptions, give rises to new generalized Einstein metric conditions on M . We investigate hypersurfaces satisfying such curvature conditions.

On curvature characterizations of some hypersurfaces in spaces of constant curvature

“…Examples of hypersurfaces satisfying (1.4) with ρ = 0 are presented [10] (see also [14]). It is known that such hypersurfaces are Ricci-pseudosymmetric (see e.g., [5]).…”

“…We refer to [3] for a survey of results related to manifolds, and in particular to hypersurfaces satisfying pseudosymmetry type conditions. Hypersurfaces M in N n+1 s (c), n 4, having the tensor R · C expressed by a linear combination of the tensors Q(S, R), Q(g, R) and Q(S, G), were investigated in [16] (see also [10]). Among other things in [16] it was shown that such hypersurfaces M satisfy (1.5) on U H ⊂ M and in a consequence, they are Ricci-pseudosymmetric.…”

“…We also recall that semisymmetric manifolds, R · R = 0 on M [22], form a subclass of the class of the Ricci-semisymmetric manifolds. For precise definitions of the symbols used we refer to Sections 2 and 3 of this paper and Sections 2 and 3 of [10] (see also [2] and [16]). We mention that the problem of the equivalence of the conditions of semisymmetry (R · R = 0) and Ricci-semisymmetry (R · S = 0) on hypersurfaces in Euclidean spaces, named the SAWICZ problem of P. J. Ryan, lead to considerations of quasi-Einstein hypersurfaces (see e.g., [1], [8]).…”

Hypersurfaces in spaces of constant curvature satisfying some Ricci-type equations

“…Hypersurfaces M in N n+1 s (c), n ≥ 4, satisfying (9) on U H ⊂ M were investigated in many papers: [1], [3], [5]- [7], [9]- [13], [15], [17], [19]- [23] and [26]. These papers are also related to the P. J. Ryan problem (see e.g.…”

On Ricci-Pseudosymmetric Hypersurfaces in Space Forms

“…In the paper [4] (Theorem 3.1) it was shown that on every hypersurface M in Np +1 (c) , n > 4, the following identity is satisfied…”