Semisymmetry and Ricci-semisymmetry for hypersurfaces of semi-Riemannian space forms
Let \((M, g)\) be a semi-Riemannian manifold. \(M\) is called semisymmetric if some natural product of its Riemann curvature tensor with itself is zero, and Ricci-semisymmetric if a natural product of its Riemann curvature tensor with its Ricci curvature tensor vanishes. This paper deals with a conjecture of P. J. Ryan, according to which the two notions are equivalent for a hypersurface of Euclidean \(n\)-space. The authors note that this conjecture is false, as was recently proved by one of them; it is however known to be true in a number of special cases. In this paper, they prove that it is true for hypersurfaces of semi-Riemannian space-forms for which a natural product of the Weyl curvature tensor with the Riemann tensor is zero. This result is achieved through elaborate algebraic computations on the curvature tensors of hypersurfaces.
scientific article; zbMATH DE number 1513038