## 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

### Keywords:

semi-Euclidean space; hypersurface; semisymmetric manifold
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