The General Type of Singularity of a Set of 2n − 1 Smooth Functions of n Variables
The General Type of Singularity of a Set of 2n − 1 Smooth Functions of n Variables
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DOI：
10.1007/978-1-4612-2972-8_23
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发表时间：
1992
影响因子：
2.5
通讯作者：
H. Whitney
中科院分区：
文献类型：
--
作者：
H. Whitney
1.导论.设n-空间E”的一个区域R，或更一般地说，可微n-流形的一个区域R，可微映射到m-空间Em。如果m 2：：2n，总是可能的[1; 818]，[3]，通过稍微改变映射函数I（也让任何有限数量的导数任意地稍微改变），获得处处正则的映射f*。也就是说，对于R中的任何p，以及任何独立向量集U1，.，在p，f* 处的R中的Un将这些向量携带成独立向量。这里，向量等于”切空间”中的向量，等于微分。因此，p的某个邻域U被I一一映射。本文的目的是确定在m= 2n-1的情况下，通过f的微小变化可以得到什么。原来，任何奇点都可以变成一种固定的类型。(It将在其他论文中表明，任何光滑的n-流形都可以嵌入（2n）-空间，并且可以浸入（允许自交）（2n-I）-空间。有两个主要的定理在文件中，大致：（a）我们可以任意改变我略有，形成f*，其中奇点（点，f* 是不正规的）是孤立的，并使一定的条件（e）下面保持在每个奇点。(The自相交也可以变得简单; [3; 655，（D）]。(b)令f* 满足上述条件。然后对于任何奇点p，我们可以选择坐标系X1，...，Xn在p的邻域中，并且Y1，···，Y2 n-1在I（p）的邻域中，使得f* 由等式（4.2）精确给出。这里，f* 必须有许多导数。备注。因此，E2 n-1有一个微小的变形，它把I（U）（U是p的邻域）带入由式（4.2）给出的点集中。（B）中的变换可以大大降低f* 的类;但如果f* 是C '类或解析类，则变换也是。（a）所述的条件如下：
1. Introduction. Let a region R of n-space E", or more generally, of a differentiable n-manifold, be mapped differentiably into m-space Em. If m 2:: 2n, it is always possible [1; 818],[3], by a slight alteration of the mapping function I (letting also any finite number of derivatives change arbitrarily slightly), to obtain a mapping f* which is everywhere regular. That is, for any p in R, and any set of independent vectors Ul,..., Un in R at p, f* carries these vectors into independent vectors. Here, vector equals the vector in" tangent space" equals the differential. As a consequence, some neighborhood U of p is mapped by I in a one-one way. The object of this paper is to determine what can be obtained by slight alterations of f in case m= 2n-1. It turns out that any singularities may be made into a fixed kind.(It will be shown in other papers that any smooth n-manifold may be imbedded in (2n)-space, and may be immersed (self-intersections allowed) in (2n-I)-space.) There are two main theorems in the paper, roughly:(a) We may alter I arbitrarily slightly, forming f*, for which the singular points (points where f* is not regular) are isolated, and such that a certain condition (e) below holds at each singular point.(The self-intersection may also be made simple; cf.[3; 655,(D)].)(b) Let f* satisfy the condition mentioned. Then for any singular point p, we may choose coordinate systems Xl,..•, Xn in a neighborhood of p and Yl,•••, Y2n-1 in a neighborhood of I (p) such that f* is given exactly by the equations (4.2). Here, f* must have many derivatives. Remark. As a consequence, there is a slight deformation of E2n-1 which carries I (U)(U a neighborhood of p) into the set of points given by (4.2). The transformations in (b) may lower the class of f* considerably; but if f* is of class C', or analytic, the transformations will be also. The condition mentioned in (a) is the following: