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On some class of hypersurfaces with three distinct principal curvatures. (English) Zbl 1091.53010

Opozda, Barbara (ed.) et al., PDEs, submanifolds and affine differential geometry. Proceedings of the conference and autumn school, Bȩdlewo, Poland, September 23–27, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 69, 145-156 (2005).
A semi-Riemannian manifold \((M,g)\) is said to be pseudo-symmetric if \(R\cdot R\) and \(Q(g,R)\) are linearly dependent at every point of the manifold. The condition is a generalization of \(R\cdot R=0\) (semi-symmetric spaces) and arose during the study of totally umbilical submanifolds of semi-symmetric manifolds. It also appears when considering hypersurfaces of Euclidean spaces. A semi-Riemannian manifold \((M,g), n\geq 4\) is called a manifold with pseudo-symmetric Weyl tensor if at every point of \(M,\) the tensors \(C\cdot C\) and \(Q(g, C)\) are linearly dependent. This is equivalent on \({\mathcal U}_C=\{x \in M\mid C\not = 0\) at \(x\}\) to \(C\cdot C=L_CQ(g,C),\) where \(L_C\) is some function on \({\mathcal U}_C.\) In the present paper, hypersurfaces in spaces of constant curvature with some special minimal polynomial of the second fundamental tensor \(H\) of third degree are considered and studied. It is shown that such a hypersurface is pseudo-symmetric if it is a manifold with pseudo-symmetric Weyl tensor. Hypersurfaces with three distinct principal curvatures are characterized.
For the entire collection see [Zbl 1077.53002].

MSC:

53B25 Local submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B20 Local Riemannian geometry
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