论文标题
切线射线叶及其相关的外台球
Tangent ray foliations and their associated outer billiards
论文作者
论文摘要
让$ v $成为一个完整的，脐带（但不是完全测量）的超出表面$ n $的单位矢量字段；例如，在单位球上$ s^{2k-1} \ subset \ mathbb {r}^{2k} $，或在双曲线空间中的housphere上。我们为带有初始速度$ v $（和$ v $）的光线的$ v $提供了必要的条件，以三$ $ u $ $ u $ $ u $。我们发现并探索这些向量领域，地球矢量字段和$ n $的联系结构之间的关系。 当射线与$ \ pm v $ foliate $ u $相对应时，$ v $引起了台球表为$ u $的外台球地图。我们在$ n $上描述了相关外台球图的单位矢量字段。另外，我们详细研究了一个特定的示例，即，当$ n \ simeq \ mathbb {r}^3 $是四维双曲线空间的housosphere，而$ v $是$ n $上的单位向量字段，通过使$ s^{3} $的Hopf vector Field的立体投影获得了$ n $。在相应的外台球图中，我们发现明确的周期轨道，无界轨道和有界的非周期轨道。最后，我们以几个有关双分裂矢量场及其相关外台球动态的拓扑和几何形状的问题。
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.
