Tangent ray foliations and their associated outer billiards

Abstract

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.