A classification of immersions of the two-sphere
A classification of immersions of the two-sphere
复制标题
二球体浸没的分类
DOI：
10.1090/s0002-9947-1959-0104227-9
复制
发表时间：
1959
影响因子：
1.3
通讯作者：
S. Smale
中科院分区：
文献类型：
--
作者：
S. Smale
一个 C' 可微流形在另一个中的浸入是第一个到第二个的正则映射（雅可比行列式具有最大秩的 C' 映射）。如果浸入的同伦在每个阶段都是规则的并且切丛的诱导同伦是连续的，则该浸入的同伦被称为规则的。人们对正则同伦下浸没分类的一般问题知之甚少。 Whitney [5] 表明，n 维流形中的 k 维流形的两个浸入式（n>2k+2）通常是同伦的，当且仅当它们是同伦的。 Whitney-Graustein 定理 [4] 对圆 S 在平面 E2 中的浸没进行分类。在我的论文 [3] 中，这个定理被扩展到 E2 被任何 C2 流形 Mn 代替，n> 1 的情况。据我所知，这些是唯一已知的结果。在本文中，我们给出了欧几里得 n 空间 E 中 2 球体 S2 的浸没分类，n>2，相对于正则同伦。令 V.,2 为 En 中所有 2 帧的 Stiefel 流形。如果f和g是S2在En中的两次浸入，则定义了不变量Q(f,g)Cw2(Vn,2)。
An immersion of one C' differentiable manifold in another is a regular map (a C' map whose Jacobian is of maximum rank) of the first into the second. A homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous. Little is known about the general problem of classification of immersions under regular homotopy. Whitney [5] has shown that two immersions of a k-dimensional manifold in an n-dimensional manifold, n>2k+2, are regularly homotopic if and only if they are homotopic. The Whitney-Graustein Theorem [4] classifies immersions of the circle S in the plane E2. In my thesis [3] this theorem is extended to the case where E2 is replaced byany C2 manifold Mn, n> 1. As far as I know, these are the only known results. In this paper we give a classification of immersions of the 2-sphere S2 in Euclidean n-space E , n>2, with respect to regular homotopy. Let V.,2 be the Stiefel manifold of all 2-frames in En. If f and g are two immersions of S2 in En, an invariant Q(f, g) Cw2(Vn,2) is defined.