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Let $v$ be a unit vector field on a complete, umbilic (but not totallygeodesic) 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. Wegive necessary and sufficient conditions on $v$ for the rays with initialvelocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find andexplore relationships among these vector fields, geodesic vector fields, andcontact structures on $N$.  When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces anouter billiard map whose billiard table is $U$. We describe the unit vectorfields on $N$ whose associated outer billiard map is volume preserving. Also westudy a particular example in detail, namely, when $N \simeq \mathbb{R}^3$ is ahorosphere of the four-dimensional hyperbolic space and $v$ is the unit vectorfield on $N$ obtained by normalizing the stereographic projection of a Hopfvector field on $S^{3}$. In the corresponding outer billiard map we findexplicit periodic orbits, unbounded orbits, and bounded nonperiodic orbits. Weconclude with several questions regarding the topology and geometry ofbifoliating vector fields and the dynamics of their associated outer billiards.

Tangent ray foliations and their associated outer billiards