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Semisymmetry and Ricci-semisymmetry for hypersurfaces of semi-Euclidean spaces. (English) Zbl 0951.53015

Let \(R\) be the Riemann-Christoffel curvature tensor, \(S\) be the Einstein curvature tensor, and \(C\) be the Weyl conformal curvature tensor of a manifold. In this paper, P. J. Ryan’s problem on the equivalence of the conditions \(R\cdot R=0\) and \(R\cdot S=0\) for hypersurfaces is considered. As the main result, it is proved that the above conditions are equivalent for general hypersurfaces of semi-Euclidean spaces in any dimension, if these hypersurfaces satisfy the curvature condition \(C\cdot R=0\).

MSC:

53B25 Local submanifolds
53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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