切射线叶理及其相关的外球
Tangent ray foliations and their associated outer billiards
摘要：
令$v$是空间形式中完备脐点（但不是全测地线）超曲面$N$上的单位向量场;例如单位球面$S^{2k-1} \subset \mathbb{R}^{2k}$上，或双曲空间中的时球面上。给出了以$v$（和$-v$）为初速度的射线在$v$上使$N$的外$U$成叶的充要条件.我们发现和探索这些向量场，测地线向量场，接触结构之间的关系$N$。当对应于每个$\pm v$的射线使$U$分层时，$v$诱导出其台球桌为$U$的外部台球图。我们描述了$N$上的单位向量场，其相关的外部台球映射是体积保持的。本文还详细研究了一个特例，即当$N \simeq \mathbb{R}^3$是四维双曲空间的时球，$v$是由一个Hopf向量场在$S^{3}$上的球极平面投影归一化得到的$N$上的单位向量场.在相应的外台球映射中，我们发现了显式的周期轨道，无界轨道和有界非周期轨道。最后，我们与几个问题的拓扑结构和几何形状的bifoliating向量场和动力学的相关外台球。
英文摘要：
Let $v$ be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} \subset \mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields, geodesic vector fields, and contact structures on $N$. When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. We describe the unit vector fields on $N$ whose associated outer billiard map is volume preserving. Also we study a particular example in detail, namely, when $N \simeq \mathbb{R}^3$ is a horosphere of the four-dimensional hyperbolic space and $v$ is the unit vector field on $N$ obtained by normalizing the stereographic projection of a Hopf vector field on $S^{3}$. In the corresponding outer billiard map we find explicit periodic orbits, unbounded orbits, and bounded nonperiodic orbits. We conclude with several questions regarding the topology and geometry of bifoliating vector fields and the dynamics of their associated outer billiards.
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