P.J. Ryan's problem in semi-Riemannian space forms

Abstract

A semi-Riemannian manifold \((M,g)\), \(\dim M\geq 3\), is called semisymmetric if \(R\cdot R= 0\), and Ricci-semisymmetric if \(R\cdot S= 0\), where \(R\) and \(S\) are the curvature tensor and Ricci tensor, respectively, subjected to the curvature operator \(R\). It is known that the first condition implies the second one. \textit{R. Deszcz} [Bull. Soc. Math. Belg., Ser. A 44, No. 1, 1-34 (1992; Zbl 0808.53012)] showed that the converse is not true. \textit{P. J. Ryan} [Colloq. Math. 26, 175-182 (1972; Zbl 0243.53028)] posed the question of whether these two coditions are equivalent for hypersurfaces of Euclidean space \(E^{n+1}\). The affirmative answer is established for \(n= 4\) by the authors of the present paper [Kyungpook Math. J. 37, 371-376 (1997; Zbl 0896.53024)] and for Lorentzian hypersurfaces of a Minkowski space by \textit{F. Defever}, \textit{R. Deszcz} and \textit{L. Verstraelen} [Acta Math. Hung. 74, No. 1-2, 7-17 (1997; Zbl 0922.53009)]. Now, this result is generalized to the case of nondegenerate hypersurfaces of a 5-dimensional semi-Riemannian space form \(N^5(c)\).