Topology of Dupin hypersurfaces with six distinct principal curvatures

Abstract

Let M be a Dupin hypersurface in the unit sphere S n+1 with six distinct principal curvatures. We will prove in the present paper that M is either diffeomorphic to SU (2) × SU (2)/Q 8 or homeomorphic to a tube around an embedded 5-dimensional complex Fermat hypersurface X 5 (2) in S 13 , where Q 8 ⊂ SU (2) = Sp(1) denotes the subgroup {±1, ±i, ±j, ±k} and X 5 (2) = {[z 0 , z 1 , · · · z 6 ] ∈ CP 6 |z 2 0 + z 2 1 + · · ·+ z 2 6 = 0}. Moreover, in the former case, all of the focal manifolds are diffeomorphic to S 3 × RP 2 ; In the latter case, one of the focal manifolds is homeomorphic to X 5 (2).

The Isoparametric Story, a Heritage of Élie Cartan

“…Fang also showed [28] that, when g = 6, the isoparametric hypersurface is diffeomorphic (respectively, homotopic) to the homogeneous example when the equal multiplicity is 1 (respectively, 2); the statement in fact holds true in the more general proper Dupin category.…”

Compact Dupin Hypersurfaces

“…Grove and Halperin [28] also gave a list of the integral homology of all compact proper Dupin hypersurfaces, and Fang [26] found results on the topology of compact proper Dupin hypersurfaces with g = 6 principal curvatures.…”

Riemannian Submanifolds: A Survey

“…Thus, these focal submanifolds are two noncongruent minimal taut homogeneous embeddings of RP 2 × S 3 in S 7 . Fang (1995) studied the topology of isoparametric hypersurfaces with six principal curvatures. Peng and Hou (1989) gave explicit forms for the isoparametric polynomials of degree six for the homogeneous isoparametric hypersurfaces with g = 6.…”