  Let $M_{\kappa }$ be the three-dimensional space form of constant curvature
$\kappa =0,1,-1$, that is, Euclidean space $\mathbb{R}^{3}$, the sphere $S^{3}
$, or hyperbolic space $H^{3}$. Let $S$ be a smooth, closed, strictly convex
surface in $M_{\kappa }$. We define an outer billiard map $B$ on the four
dimensional space $\mathcal{G}_{\kappa }$ of oriented complete geodesics of
$M_{\kappa }$, for which the billiard table is the subset of
$\mathcal{G}_{\kappa }$ consisting of all oriented geodesics not intersecting
$S$. We show that $B$ is a diffeomorphism when $S$ is quadratically convex.
  For $\kappa =1,-1$, $\mathcal{G}_{\kappa }$ has a K\"{a}hler structure
associated with the Killing form of $\operatorname{Iso}(M_{\kappa })$. We prove
that $B$ is a symplectomorphism with respect to its fundamental form and that
$B$ can be obtained as an analogue to the construction of Tabachnikov of the
outer billiard in $\mathbb{R}^{2n}$ defined in terms of the standard symplectic
structure. We show that $B$ does not preserve the fundamental symplectic form
on $\mathcal{G}_{\kappa }$ associated with the cross product on $M_{\kappa }$,
for $\kappa =0,1,-1$.
  We initiate the dynamical study of this outer billiard in the hyperbolic case
by introducing and discussing a notion of holonomy for periodic points.
