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Đề tài " On the classi_cation of isoparametric hypersurfaces with four distinct principal curvatures in spheres "

In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m1 , m2 of the principal curvatures satisfy m2 ≥ 2m1 − 1. This inequality is satisfied for all but five possible pairs (m1 , m2 ) with m1 ≤ m2 . | Annals of Mathematics On the classi_cation of isoparametric hypersurfaces with four distinct principal curvatures in spheres By Stefan Immervoll Annals of Mathematics 168 2008 1011-1024 On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres By Stefan Immervoll Abstract In this paper we give a new proof for the classification result in 3 . We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m1 m2 of the principal curvatures satisfy m2 2m1 1. This inequality is satisfied for all but five possible pairs m1 m2 with m1 m2. Our proof implies that for m1 m2 1 1 the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs m1 m2 with m1 m2 see 13 1 and 15 this stronger form of our result is incorrect for the three pairs 3 4 6 9 and 7 8 there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds and for the two pairs 2 2 and 4 5 there exist homogeneous examples that are not of Clifford type cf. 5 4.3 4.4 . 1. Introduction In this paper we present a new proof for the following classification result in 3 . Theorem 1.1. An isoparametric hypersurface with four distinct principal curvatures in a sphere is of Clifford type provided that the multiplicities m1 m2 of the principal curvatures satisfy the inequality m2 2m1 1. An isoparametric hypersurface M in a sphere is a compact connected smooth hypersurface in the unit sphere of the Euclidean vector space V RdimV such that the principal curvatures are the same at every point. By 12 Satz 1 the distinct principal curvatures have at most two different multiplicities m1 m2 . In the following we assume that M has four distinct principal curvatures. Then the only possible pairs m1 m2 with m1