The splitting theorem for orbifolds
The splitting theorem for orbifolds
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轨道折叠定理
DOI：
10.1215/ijm/1256060999
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发表时间：
1994
影响因子：
0.6
通讯作者：
Shunhui Zhu
中科院分区：
文献类型：
--
作者：
Joseph E. Borzellino;Shunhui Zhu
本文研究Cheeger-Gromoll [CG]分裂定理在黎曼轨道上的推广。粗略地说，黎曼轨道折叠是一个度量空间，局部地由有限等距群在黎曼流形的等价物上建模。Orbifold一词是由W. Thurston [T]大约在1976-77年的某个时候。这个术语的意思是指流形上的群作用的轨道空间。一个类似的概念是由I。佐竹在1956年，他使用的术语V-流形（见[S1]）。“V”意味着一个类似圆锥的奇点。从那时起，orbifold就成为了首选术语。回想一下，如果M是一个包含一条直线的具有非负Ricci曲率的完全连通n维黎曼流形，那么Cheeger- Gromoll分裂定理[CG]指出M等距于N × R。回想一下，直线是单位速度测地线γ：R → M，使得对于任何s，t ∈ R，d（γ（s），γ（t））=| s− t|.
In this paper we wish to examine a generalization of the splitting theorem of Cheeger–Gromoll [CG] to Riemannian orbifolds. Roughly speaking, a Riemannian orbifold is a metric space locally modelled on quotients of Riemannian manifolds by finite groups of isometries. The term orbifold was coined by W. Thurston [T] sometime around the year 1976–77. The term is meant to suggest the orbit space of a group action on a manifold. A similar concept was introduced by I. Satake in 1956, where he used the term V– manifold (See [S1]). The “V” was meant to suggest a cone–like singularity. Since then, orbifold has become the preferred terminology. Recall that if M is a complete connected n–dimensional Riemannian manifold with nonnegative Ricci curvature that contains a line, then the Cheeger– Gromoll Splitting Theorem [CG] states that that M is isometric to N × R. Recall that a line is a unit speed geodesic γ : R → M such that for any s, t ∈ R, d(γ(s), γ(t)) = |s− t|.