ON SOME TYPES OF ISOPARAMETRIC HYPERSURFACES IN SPHERES I
ON SOME TYPES OF ISOPARAMETRIC HYPERSURFACES IN SPHERES I
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DOI：
10.2748/tmj/1178240877
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发表时间：
1975
影响因子：
0.5
通讯作者：
H. Ozeki;M. Takeuchi
中科院分区：
文献类型：
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作者：
H. Ozeki;M. Takeuchi
介绍。本文是第一部分[13]的延续。在本文的前半部分，我们研究了球体中的齐次等参超曲面。根据 Hsiang-Lawson [8]，球体中的每个齐次超表面都表示为 2 阶黎曼对称空间的线性各向同性群的轨道。在第 1 节中，我们研究黎曼对称空间及其轨道的线性各向同性表示。 §2 和 §3 致力于研究齐次等参超曲面、它们的分类和不变多项式。在第 4 节和第 5 节中，我们为球体中的每个齐次等参超曲面显式构造了定义多项式 F，这是由 Cartan [3] 在 g - 3 的情况下完成的。在后半部分，我们证明了在 g = 4 且 m^ 或 w2 = 2 的情况下球体中的每个闭等参超曲面都是齐次的。 Cartan [4] 指出，在没有证明的情况下，在 g = 4 的情况下，具有相同重数的球体中的每个闭等参超曲面都是齐次的。如果 m^ = m2 = 2，我们在第 9 节中给出了其证明的简要概述。在第 6 节中，我们针对一些同质示例展示了 {pa, qa] 的显式形式。我们看到，对于 g = 4、m^ — 4 和 m2 = 3 的齐次等参超曲面，其定义多项式 — F 不满足第一部分第 6 节中给出的条件 (B)。因此可以得出结论，我们在第一部分定理 2 中为 F = H 和 r = 1 构造的示例不是齐次的。因此，S 中至少存在两种具有相同重数的等参超曲面；一种是同质的，另一种则不是。寻找局部几何量来区分它们似乎是一个有趣的问题。
Introduction. This paper is a continuation of Part I [13]. In the first half of the present paper, we study the homogeneous isoparametric hyper surf aces in spheres. Every homogeneous hyper surf ace in a sphere is represented as an orbit of a linear isotropy group of a Riemannian symmetric space of rank 2, due to Hsiang-Lawson [8]. In §1, we study the linear isotropy representations of Riemannian symmetric spaces and their orbits in general. §2 and §3 are devoted to a study of the homogeneous isoparametric hyper surf aces, their classification and invariant polynomials. In § 4 and § 5, we construct explicitly the defining polynomial F for each homogeneous isoparametric hypersurface in a sphere, which was done by Cartan [3] in case g — 3. In the second half, we prove that every closed isoparametric hypersurface in a sphere in case g = 4 and m^ or w2 = 2 is homogeneous. Cartan [4] indicated, without proof, that in case g = 4, every closed isoparametric hypersurface in a sphere with the same multiplicities is homogeneous. In case m^ = m2 = 2, we give a brief outline of its proof in §9. In §6, we exhibit explicit forms of {pa, qa] for some of the homogeneous examples. We see that, for a homogeneous isoparametric hypersurface with g = 4, m^ — 4 and m2 = 3, its defining polynomial — F does not satisfy the condition (B) given in § 6 of Part I. Thus one can conclude that our example constructed in Theorem 2 of Part I for F = H and r = 1 is not homogeneous. Consequently, there are at least two types of isoparametric hypersurfaces in S with the same multiplicities; one is homogeneous, and the other is not. It seems to be an interesting problem to seek a local geometric quantity in order to distinguish them.