Unique Triangulation of the Orbit Space of a Differentiable Transformation Group and its Applications
Unique Triangulation of the Orbit Space of a Differentiable Transformation Group and its Applications
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可微变换群轨道空间的唯一三角剖分及其应用
DOI：
10.2969/aspm/00910041
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发表时间：
1987
影响因子：
2.4
通讯作者：
M. Shiota
中科院分区：
文献类型：
--
作者：
T. Matumoto;M. Shiota
设G是紧李群.我们考虑一类Ck维m的仿紧可微流形M，它具有一个C\类的可微G-作用G X M----+M，我们称之为Ck G-流形.我们将看到一个可微的（即，C k，其中I：：k 'k < 00）G-流形M是等变同构于一个真实的解析的（即，C“）G-流形（定理1.3）.一个CO 1等变光滑是“唯一”确定的（定理1.2- 1.2 '）：唯一的直到C”的等变光滑，如果M是紧的或更一般地M只有有限个轨道类型，并且唯一的直到次解析的C1等变光滑。本文利用有限维线性表示空间中具有有限轨道型的真实的解析G-流形的等变嵌入定理（定理1.1）。回顾了Hironaka [HI]在第二节中定义的亚解析集和映射的概念，我们在第三节中讨论了真实的解析G-流形。在轨道空间上引入一个自然的次解析集结构（定理3.1），按轨道类型过滤的分层是次解析的（引理3.2）。因此，我们有一个唯一的轨道空间的三角剖分，它与次解析集结构相容，因此与轨道型分解相容，在这个意义上，两个这样的三角剖分有一个共同的次解析和组合细分（定理3.3），使用[SY]的结果。结合这些结果，我们得到了一个唯一的三角形的轨道空间也对任何可微G-流形M。注意到带边界的可微G-流形M的轨道空间只不过是可微GXZ 2-流形DM的轨道空间，其中DM是M的二重流形;
Let G be a compact Lie group throughout this paper. We consider a paracompact differentiable manifold M of class C k and dimension m with a differentiable G-action G X M----+M of class C\ which we call a Ck G-manifold. We shall see that a differentiable (i.e., C k with I ::;'k< 00) G-manifold Mis equivariantly diffeomorphic to a real analytic (i.e., C"') G-manifold (Theorem 1.3). A COl equivariant smoothing is "uniquely" determined (Theorems 1.2-1.2'): unique up to C" equivariant diffeomorphism if M is compact or more generally M has only a finite number of orbit types and unique up to subanalytic C1 equivariant diffeomorphism in general. We use here the equivariant embedding theorem for real analytic G-manifold with finite orbit types in a finite dimensional linear representation space (Theorem 1.1). Reviewing the notion of subanaIytic sets and maps defined by Hironaka [HI] in Section 2, we treat the real analytic G-manifolds in Section 3. A natural subanalytic set structure is introduced on the orbit space (Theorem 3.1) and the stratification filtered by orbit types is subanalytic (Lemma 3.2). So, we have a unique triangulation of the orbit space which is compatible with the subanalytic set structure and consequently with the orbit type decomposition in the sense that two such triangulations have a common subanalytic and combinatorial subdivision (Theorem 3.3) using the results of [SY]. Combining these results we get a unique triangulation of the orbit space also for any differentiable G-manifold M. Notice that the orbit space of a differentiable G-manifold M with boundary is nothing but that of the differentiable GXZ2-manifold DM, where DMis the double of M;