On the topology of isoparametric hypersurfaces with four distinct principal curvatures期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Authors:	Fuquan Fang

Institution:	Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China

Abstract:	Let be the pair of multiplicities of an isoparametric hypersurface in the unit sphere $S^{n+1}$ with four distinct principal curvatures -w.r.g., we assume that $m_-\le m_+$ . In the present paper we prove that, in the case 4B2 of U. Abresch (Math. Ann. 264 (1983), 283-302) (i.e., where ), must be either 2 or 4. As a by-product, we prove that the focal manifold of an isoparametric hypersurface is homeomorphic to a $S^{m_+}$ bundle over $S^{m_++m_-}$ if one of the following conditions holds: (1) and or $7\pmod{8}$ ; (2) and $m_+=0\pmod{4}$ . This generalizes partial results of Wang (1988) about the topology of Clifford type examples. Consequently, the hypersurface is homeomorphic to an iterated sphere bundle under the above condition.

Keywords:	Isoparametric hypersurface principal curvature multiplicity of principal curvature iterated sphere bundle

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