Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability
Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability
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C ∞ 映射的稳定性：II. 无穷小稳定性意味着稳定性
DOI：
10.2307/1970668
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发表时间：
1969
影响因子：
4.9
通讯作者：
J. Mather
中科院分区：
文献类型：
--
作者：
J. Mather
本文的目的是证明一个有限维Co流形到另一个有限维Co流形的适当的无穷小稳定映射是稳定的。这一结果被称为定理1？3.我们使用的方法来证明这一结果证明更多;两件事，它证明是作为定理2和3在？3.这个定理的意义如下。稳定映射在可微映射的奇点理论中起着重要的作用（参见[8][9]）。因此，需要有方法来证明特定映射是稳定的。本文的结果是朝着这样的方法迈进了一步，因为一般来说证明一个映射f是无穷小稳定的比证明f是稳定的更容易。本文的组织结构如下。在哪？1.我们确定我们的符号和术语。所有这些都是非常标准的，除了“流形”有角。然而，似乎最好是明确，因为术语的确切含义因作者而异。在哪？2.在一个流形到另一个流形的Co映射空间上引入了拓扑W = WO。主要结果（？3)都是按照这个拓扑结构来表达的。一般来说，复合相对于拓扑W是不连续的。一个主要部分？2致力于证明复合映射的各种限制是连续的。在哪？第三部分，主要研究结果。注意，定理3平凡地蕴涵定理1和定理2。我们将参考本系列中的前一篇论文[5]作为I。在哪？4、应用I.命题1基本上是一个重述的“分工定理”方面的拓扑结构介绍？2.另一方面，命题2引入了一些新的东西，因为它扩大了我们可以“除”的函数集。在哪？5，我们解释了塞利的结果的拓扑，我们
The purpose of this paper is to prove that a proper, infinitesimally stable mapping of one finite dimensional Co manifold into another is stable. This result is stated as Theorem 1 of ? 3. The method we use to show this result proves somewhat more; two things which it proves are stated as Theorems 2 and 3 in ? 3. The interest of this theorem is the following. Stable mappings play an important role in the theory of singularities of differentiable mappings (cf. [8] and [9]). Therefore it is desirable to have methods for proving that particular mappings are stable. The result of this paper is a step towards such methods, since it is generally easier to prove that a mapping f is infinitesimally stable than it is to prove that f is stable. This paper is organized as follows. In ? 1, we set down our notation and terminology. All of this is quite standard except that "manifolds" have corners. However it seemed desirable to be explicit, since the precise meaning of the terminology varies from author to author. In ? 2, we introduce the topology W = WO, on the space of Co mappings of one manifold into another. The main results (? 3) are formulated in terms of this topology. In general, composition is not continuous with respect to the topology W. A major part of ? 2 is devoted to proving that various restrictions of the composition mapping are continuous. In ? 3, we state the main results. Note that Theorem 3 trivially implies Theorems 1 and 2. We will refer to the previous paper [5] in this series as I. In ? 4, we apply the results of I. Proposition 1 is essentially a restatement of the "division theorem" in terms of topologies introduced in ? 2. Proposition 2, on the other hand, introduces something new, in that it enlarges the set of functions that we can "divide" by. In ? 5, we interpret a result of Seeley in terms of the topologies that we