Isoparametric hypersurfaces with four or six distinct principal curvatures
Isoparametric hypersurfaces with four or six distinct principal curvatures
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DOI：
10.1007/bf01459125
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发表时间：
1983-09
影响因子：
1.4
通讯作者：
U. Abresch
中科院分区：
文献类型：
--
作者：
U. Abresch
等参超曲面，即具有恒定主曲率的超曲面，在某种意义上是超曲面理论的最简单的例子；然而，球面中等参超曲面的完全分类还没有实现。困难的原因在于，与欧几里得或双曲空间中的情形相比，不同主曲率的个数g可以大于2。Cartan首先考虑了等参超曲面，解决了Ge{1，2，3}[Car]的分类问题。在研究低上齐度极小曲面时，Hsiang和Lawson指出，任何齐次等参超曲面一定源于某一秩为2的对称空间的各向同性表示[HSI]。Takagi和Takahashi随后详细地研究了这些例子的几何[TAK]，并注意到有两种情况是0=6，几种情况是9=4。相当多的g=4的等参超曲面--其中许多是非齐次的--可以从Clifford代数[FKM]的正交表示中得到。Mtinzner[M~N，MON]已经建立了必要条件；根据Cartan的结果，问题归结为研究g=4和g=6的情形。我们的目的是证明所有具有4或6个不同主曲率的等参超曲面的一些性质。我们指出，根据[MON]那里
Isoparametric hypersurfaces, ie hypersurfaces with constant principal curvatures, are in some sense the simplest examples for the theory of hypersurfaces; however, a complete classification of isoparametric hypersurfaces in spheres has not been achieved. The difficulties arise from the fact that in contrast to the situation in euclidean or in hyperbolic space the number g of distinct principal curvatures can be greater than 2.Cartan, who considered isoparametric hypersurfaces first, has solved the classification problem in case ge {1, 2, 3}[CAR]. When studying minimal surfaces of low cohomogeneity, Hsiang and Lawson remarked that any homogeneous isoparametric hypersurface must stem from the isotropy representation of some symmetric space of rank2 [HSI]. Takagi and Takahashi then studied the geometry of these examples in detail [TAK] and noticed that there occur two cases with 0= 6 and several cases with 9= 4. Quite a mass of isoparametric hypersurfaces with g= 4-many of them are non-homogeneous-can be obtained from orthogonal representations of Clifford algebras [FKM]. Necessary conditions have been established by Mtinzner [M~ N, MON 2]; in view of Cartan's results the problem is reduced to studying the cases g= 4 and g= 6. It is our aim to prove some properties of all isoparametric hypersurfaces with 4 or 6 distinct principal curvatures. We point out that according to [MON] there