切片定理与无穷维轨道型分层
Slice theorem and orbit type stratification in infinite dimensions
摘要：
建立了局部凸流形上局部凸李群作用的一般切片定理，将Palais的经典切片定理推广到无限维.我们讨论了两个重要的设置下，这个定理的假设得到满足。首先，利用Glöckner的反函数定理，我们证明了紧李群在Fréchet空间上的线性作用允许一个切片。其次，利用Nash-Moser定理，建立了驯服Fréchet流形上驯服Fréchet李群的驯服作用的切片定理。为此目的，我们开发的概念，分次黎曼度量，它允许建设的路径长度度量兼容的流形拓扑结构和本地除了。最后，我们推广了有限维空间中的一个经典结果，证明了切片的存在意味着流形分解为群作用的轨道型是一个分层。
英文摘要：
We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash--Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.
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