Geodesic foliations by circles
Geodesic foliations by circles
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按圆圈测地线叶状
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发表时间：
1975
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通讯作者：
A. Wadsley
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作者：
A. Wadsley
D. B. A. Epstein 在论文 [2] 中对紧凑三流形圆的平滑叶理进行了完整分析。本质上，他证明了所有这些叶状结构都是通过平滑圆周作用的轨道对流形进行分解而出现的。本文的定理表明，对于任意光滑流形（无论是否紧致），以及满足一定（相当强）正则性条件的圆叶状结构，情况也是如此。众所周知，并非所有圆的叶状结构都是作为 S 的某些作用的轨道而出现的，论文 [2] 提出了叶状非紧三流形作为这一命题的反例。然而，在维度大于三的叶状紧流形的情况下是否存在这样的例子是一个悬而未决的问题。 C 流形 M 上的 C 流是 M 上的加性实数的 C 作用 μ: R X M —• M。没有不动点的 C 流，其每个轨道都是紧的，产生了流形的 C 叶状圆。此外，任何由流形 M 的圆形成的 C 叶状结构都会在 M（的双覆盖）上产生 C 流。这里提出的定理版本是针对流动而言的，根据微分形式的圆叶状结构的等效版本很容易获得（参见§ 2）。定理如下。定理。设 μ: R X M —> M 为实数加法群的 C 作用 (3 < r < oo)，每个轨道均为圆，且 M 为 C 流形。那么存在一个与 μ 具有相同轨道的 C 动作 p\ S X M —> M，当且仅当 M 上存在某种黎曼度量，相对于该度量，μ 的轨道被嵌入为 M 的完全测地子流形。给定 M 上的圆动作，找到一些这样的度量很容易（参见第 3 节），反之证明则需要更多的努力。作者要感谢大卫·爱泼斯坦的温柔鼓励和许多有益的建议。
Smooth foliations by circles of compact three-manifolds have been completely analysed by D. B. A. Epstein in the paper [2]. Essentially, he shows that all such foliations arise as a decomposition of the manifold by the orbits of a smooth circle action. The theorem of this paper shows that the same is true of an arbitary smooth manifold, compact or not, with a foliation by circles satisfying a certain (rather strong) regularity condition. It is known that not all foliations by circles arise as the orbits of some action by S indeed, the paper [2] presents a foliated noncompact three-manifold as a counter-example to such a proposition. However, it is an open question whether or not such examples exist in the case of a foliated compact manifold of dimension greater than three. A C flow on a C manifold M is a C action μ: R X M —• M of the additive reals on M. A C flow without fixed points, each of whose orbits is compact, gives rise to a C foliation of the manifold by circles. Further, any C foliation by circles of a manifold M gives rise to a C flow on (a double cover of) M. The version of the theorem presented here is stated for flows an equivalent version for circle foliations in terms of differential forms is readily obtainable (see § 2). The theorem is the following. Theorem. Let μ: R X M —> M be a C action (3 < r < oo) of the additive group of real numbers with every orbit a circle, and M a C manifold. Then there is a C action p\ S X M —> M with the same orbits as μ if and only if there exists some riemannian metric on M with respect to which the orbits of μ are embedded as totally geodesic submanifolds of M. Finding some such metric given a circle action on M is easy (see § 3) the proof of the converse requires a little more effort. The author wishes to thank David Epstein for his gentle encouragement and for his many helpful suggestions.