Stability of C ∞ mappings: VI the nice dimensions
Stability of C ∞ mappings: VI the nice dimensions
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C ∞ 映射的稳定性：VI 的良好维度
DOI：
10.1007/bfb0066824
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发表时间：
1971
期刊：
影响因子：
--
通讯作者：
J. Mather
中科院分区：
文献类型：
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作者：
J. Mather
A smooth（乐）. COO）映射f从一个有限维流形到另一个有限维流形被称为是如果f的任何足够接近的逼近g等价于f，在这个意义上，存在源和目标到它们自身的光滑同态，它们将f变换为g。这是我们在本系列论文中研究的基本概念[3J]。(See在这个系列中，II给出了稳定映射的精确定义。）在V中，我们给出了稳定映射的一个刻划，它表明至少对于真映射，映射是否稳定仅取决于映射的局部性质.另一方面，人们可以找到不合适的映射，并且不稳定（本质上是病态的原因），即使它们满足局部稳定性标准。由于这个原因，在非正常情况下的”稳定映射“的概念没有什么意义。因此，我们不得不提出以下问题。设和pP分别是n维和p维流形. N到P的真稳定映射在N到P的真映射集中稠密吗？
A smooth (Le. COO) mapping f from one finite dimensional manifold to another is said to be if any sufficiently close approximation g to f is equivalent to f in the sense that there exist smooth diffeomorphisms of the source and the target onto themselves which transform f into g. This is the basic notion which we have studied in this series of papers [3J.(See II in this series for a precise definition of stable mappings.) In V of this s eries, we gave a characterization of stable mappings, which shows that at least for proper mappings, whether a mapping is stable or not depends only on local properties of the mapping. On the other hand, one can find mappings which are not proper and fail to be stable (for essentially pathological reasons) even though they satisfy the local criteria for stability. For this reason, the notion of" stable mappingsIt in the non proper case is of little interest. Thus, we are led to formulate the following question. Let and pP be manifolds of dimensions n and p respectively.Question. Are the proper stable mappings of N into P dense in the set of proper mappings of N into P?