THE CLOSED GEODESIC PROBLEM FOR COMPACT RIEMANNIAN 2-ORBIFOLDS
THE CLOSED GEODESIC PROBLEM FOR COMPACT RIEMANNIAN 2-ORBIFOLDS
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紧凑黎曼二环折的闭合测地线问题
DOI：
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发表时间：
1996
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通讯作者：
Benjamin G. Lorica
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作者：
Joseph E. Borzellino;Benjamin G. Lorica
在本文中，我们研究了黎曼 2 轨道上光滑闭合测地线的存在性问题。粗略地说，黎曼轨道折叠是通过有限等距群对黎曼流形的商进行局部建模的度量空间。事实证明，黎曼轨道折叠继承了自然分层长度空间结构，并且局部表现良好，因此可以应用 Alexandrov 几何和几何分析技术将黎曼流形的标准结果扩展到黎曼轨道折叠。我们在本文中考虑的2-轨道折叠是其底层空间是无边界流形的轨道折叠。人们可以将这种黎曼轨道视为具有一些显着奇异锥点的 2-流形，其邻域与 2-圆盘的商等距，并通过固定圆盘中心的有限阶循环群进行度量。我们考虑的 2-orbifolds 分为两类，我们将使用不同的技术来处理。第一种情况是轨道折叠的底层空间是单连通的（通常的拓扑意义上），即轨道折叠的底层空间是2-球体S2。此类轨道折叠包含所有可定向的坏 2-轨道折叠的集合，即那些不作为 S2 与某个度量由一组有限的等距正确间断作用的商而出现的轨道折叠。这些糟糕的 2 环折就是通常所说的泪珠和足球的例子。第二类双环折叠是那些其底层空间不是通常意义上的简单连接的双环折叠。轨道折叠的基本参考是 [T]，而 [Bl] 则采用更多黎曼观点。我们将使用的许多关于黎曼轨道折叠的结果已以已发表的形式出现在 [B2] 中。在我们陈述和讨论黎曼轨道折叠的结果之前，我们想回顾一下用于证明 Fet 和 Lyusternik [FL] 经典定理的方法和思想：在任何紧黎曼流形上
In this paper, we examine the question of the existence of a smooth closed geodesic on Riemannian 2-orbifolds. Roughly speaking a Riemannian orbifold is a metric space locally modelled on quotients of Riemannian manifolds by finite groups of isometries. It turns out that Riemannian orbifolds inherit a natural stratified length space structure and are sufficiently well-behaved locally so that one may apply both techniques of Alexandrov geometry and geometric analysis to extend standard results about Riemannian manifolds to Riemannian orbifolds. The 2-orbifolds we consider in this paper are orbifolds whose underlying space is a manifold without boundary. One can think of such Riemannian orbifolds as 2-manifolds with some distinguished singular cone points, whose neighborhoods are isometric to a quotient of the 2-disc with some metric by a cyclic group of finite order fixing the center of the disc. The 2-orbifolds we consider fall into two categories which we will handle with different techniques. The first case is when the underlying space of the orbifold is simply connected (in the usual topological sense), that is, the underlying space of the orbifold is the 2-sphere S2. This class of orbifolds contains the set of all orientable bad 2-orbifolds, namely those that do not arise as a quotient of S2 with some metric by a finite group of isometries acting properly discontinuously. These bad 2-orbifolds are examples of what are commonly referred to as teardrops and footballs. The second class of 2-orbifolds are those whose underlying space is not simply connected in the usual sense. The basic reference for orbifolds is [T], while a more Riemannian viewpoint is taken in [Bl]. Many of the results on Riemannian orbifolds that we will use have appeared in published form in [B2]. Before we state and discuss our results for Riemannian orbifolds, we would like to recall the methods and ideas used to prove the classical theorem of Fet and Lyusternik [FL]: On any compact Riemannian manifold there