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On the topology of isoparametric hypersurfaces
with four distinct principal curvatures

Author: Fuquan Fang
Journal: Proc. Amer. Math. Soc. 127 (1999), 259-264
MSC (1991): Primary 53C40; Secondary 53B25
MathSciNet review: 1458870
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Abstract: Let $(m_-,m_+)$ 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 $3m_-=2(m_++1)$), $m_-$ must be either 2 or 4. As a by-product, we prove that the focal manifold $F_-$ of an isoparametric hypersurface is homeomorphic to a $S^{m_+}$ bundle over $S^{m_++m_-}$ if one of the following conditions holds: (1) $m_+>m_->1$ and $m_+=3,5,6$ or $7\pmod{8}$; (2) $m_+>2m_->2$ 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.

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Additional Information

Fuquan Fang
Affiliation: Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China

Keywords: Isoparametric hypersurface, principal curvature, multiplicity of principal curvature, iterated sphere bundle
Received by editor(s): June 21, 1995
Received by editor(s) in revised form: February 1, 1996, and April 30, 1997
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society

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