The Gauss Map of Surfaces in R3 and R4
The Gauss Map of Surfaces in R3 and R4
复制标题
R3 和 R4 中曲面的高斯图
DOI：
10.1112/plms/s3-50.1.27
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发表时间：
1985
影响因子：
1.8
通讯作者：
R. Osserman
中科院分区：
文献类型：
--
作者：
D. Hoffman;R. Osserman
本文代表了两条研究路线的汇合：一是当前作者 [9,10, 11] 对 R" 中曲面的广义高斯图的几何进行的研究，另一项是 Kenmotsu [13] 对 U3 中经典高斯图的研究。事实上，我们在 [11] 中早期工作的一部分是受到 Kenmotsu 的启发，Kenmotsu 非凡地发现了 U3 中具有非零平均曲率的任意曲面的表示定理（用平均曲率表示）在本文中，我们重点关注两种特别感兴趣的情况，即 U3 和 R4，我们研究了在这些情况下我们的一般结果所采用的形式，并获得了取决于相应的格拉斯曼函数的特殊形式的附加结果：R3 为球体，R4 为球体乘积。我们的方法与 Kenmotsu 的方法相同，即我们本质上利用了表面的共形结构，但不同之处在于我们主要感兴趣的是高斯图本身、它必须具有哪些属性以及它决定表面的程度。为了使这一点精确，让 S 为浸入 IR3 中的定向表面，并让 So 为通过 S 上的诱导共形结构获得的黎曼表面。因此，我们可以认为 S 由共形浸入定义。
This paper represents a confluence of two lines of investigation: one by the present authors [9, 10, 11] on the geometry of the generalized Gauss map for surfaces in R", and the other by Kenmotsu [13] on the classical Gauss map in U3. In fact, part of our earlier work in [11] was motivated by Kenmotsu's remarkable discovery of a representation theorem for arbitrary surfaces in U3 with non-vanishing mean curvature, in terms of the mean curvature function and the Gauss map of the surface. In the present paper we focus on the two cases of special interest, U3 and R4. We examine the form that our general results take in these cases, and we obtain additional results that depend on the corresponding special forms of the Grassmannian: a sphere for R3, and a product of spheres, for R4.Our approach has in common with that of Kenmotsu the fact that we make essential use of the conformal structure of the surface, while it differs in that we are interested primarily in the Gauss map itself, what properties it must possess, and the extent to which it determines the surface. To make this precise, let S be an oriented surface immersed in IR3, and let So be the Riemann surface obtained by the induced conformal structure on S. Thus we may consider S defined by a conformal immersion