# Outer billiards on the manifolds of oriented geodesics of the three
dimensional space forms

Yamile Godoy, Michael Harrison and Marcos Salvai This work was supported by
Consejo Nacional de Investigaciones Científicas y Técnicas and Secretaría de
Ciencia y Técnica de la Universidad Nacional de Córdoba.

###### Abstract

For $\kappa=0,1,-1$, let $M_{\kappa}$ be the three dimensional space form of
curvature $\kappa$, that is, $\mathbb{R}^{3}$, $S^{3}$ and hyperbolic
$3$-space $H^{3}$. Let $\mathcal{G}_{\kappa}$ be the manifold of all oriented
(unparametrized) complete geodesics of $M_{\kappa}$, i.e., $\mathcal{G}_{0}$
and $\mathcal{G}_{-1}$ consist of oriented lines and $\mathcal{G}_{1}$ of
oriented great circles.

Given a strictly convex surface $S$ of $M_{\kappa}$, we define an outer
billiard map $B_{\kappa}$ on $\mathcal{G}_{\kappa}$. The billiard table is the
set of all oriented geodesics not intersecting $S$, whose boundary can be
naturally identified with the unit tangent bundle of $S$. We show that
$B_{\kappa}$ is a diffeomorphism under the stronger condition that $S$ is
quadratically convex.

We prove that $B_{1}$ and $B_{-1}$ arise in the same manner as Tabachnikov’s
original construction of the higher dimensional outer billiard on standard
symplectic space $\left(\mathbb{R}^{2n},\omega\right)$. For that, of the two
canonical Kähler structures that each of the manifolds $\mathcal{G}_{1}$ and
$\mathcal{G}_{-1}$ admits, we consider the one induced by the Killing form of
$\operatorname{Iso}\left(M_{\kappa}\right)$. We prove that $B_{1}$ and
$B_{-1}$ are symplectomorphisms with respect to the corresponding fundamental
symplectic forms. Also, we discuss a notion of holonomy for periodic points of
$B_{-1}$.

Key words and phrases: outer billiards, space of oriented geodesics,
symplectomorphism, Jacobi field, holonomy

Mathematics Subject Classification 2020. Primary: 37C83, 53C29, 53D22.
Secondary: 53A35, 53C22, 53C35.

##  1 Introduction

The dual or outer billiard map $B$ is defined in the plane as a counterpart to
the usual inner billiards. Let $\gamma\subset\mathbb{R}^{2}$ be a smooth,
closed, strictly convex curve, and let $p$ be a point outside of $\gamma$.
There are two tangent lines to $\gamma$ through $p$; choose one of them
consistently, say, the right one from the viewpoint of $p$, and define $B(p)$
as the reflection of $p$ in the point of tangency (see Figure 1).

Figure 1: The outer billiard map in the plane

The study of the dual billiard was originally popularized by Moser [15, 16],
who considered the dual billiard map as a crude model for planetary motion and
showed that orbits of the map cannot escape to infinity. The outer billiard
map has since been studied in a number of settings; see [5, 20, 21, 23] for
surveys.

The goal of the present article is to define and study an outer billiard
system in another setting: on the space of oriented geodesics of the three-
dimensional space form $M_{\kappa}$ of constant curvature $\kappa=0,-1$ or
$1$; that is, Euclidean space $\mathbb{R}^{3}$, hyperbolic space $H^{3}$, or
the sphere $S^{3}$, respectively.

Let $\mathcal{G}_{\kappa}$ be the space of all oriented geodesics of
$M_{\kappa}$. This is a four dimensional manifold whose elements are the
oriented trajectories of complete geodesics in $M_{\kappa}$, which can also be
described as equivalence classes of unit speed geodesics
$\left[\gamma\right]$, where $\gamma\sim\sigma$ if
$\sigma\left(t\right)=\gamma\left(t+t_{o}\right)$ for some
$t_{o}\in\mathbb{R}$.

For convenience, if $\kappa=0,-1$, we sometimes call
$\mathcal{G}_{\kappa}=\mathcal{L}_{\kappa}$, the space of oriented lines of
Euclidean or hyperbolic space (which are diffeomorphic to $TS^{2}$) and
$\mathcal{G}_{1}=\mathcal{C}$, the space of oriented great circles of the
three sphere (diffeomorphic to $S^{2}\times S^{2}$).

Suppose first that $\kappa=0$ or $-1$. Let $S$ be a strictly convex compact
surface in $M_{\kappa}$, that is, for each $p\in S$, the complete totally
geodesic surface tangent to $p$ intersects $S$ only at $p$. Let

$\mathcal{U=L}_{\kappa}-\left\\{\text{oriented lines intersecting
}S\right\\}\text{.}$

The space $\mathcal{U}$ will serve as the outer billiard table; note that it
is an open submanifold of $\mathcal{L}_{\kappa}$ with boundary equal to the
three-dimensional space of lines which are tangent to $S$.

Figure 2: The outer billiard map on $\mathcal{U}$

The outer billiard map on $\mathcal{L}_{\kappa}$ associated with $S$ is the
map $B:\mathcal{U}\rightarrow\mathcal{U}$ defined as follows: Suppose that
$\ell\in\mathcal{L}_{\kappa}$ does not intersect $S$. There exist exactly two
complete totally geodesic surfaces $P_{+}$ and $P_{-}$ tangent to $S$
containing $\ell$. The orientation of $\ell$ allows us to choose the surface
on the right, say $P_{+}$ (see next paragraph). Let $p_{+}$ be the point on
$S$ tangent to $P_{+}$ and let $q_{+}$ be the point on $\ell$ realizing the
distance $d$ to $p_{+}$. Now, let $\gamma_{+}$ be the geodesic ray joining
$q_{+}$ with $p_{+}$, with $\gamma_{+}\left(0\right)=q_{+}$ and
$\gamma_{+}\left(d\right)=p_{+}$. Then $B\left(\ell\right)$ is the oriented
line obtained by parallel transporting $\ell$ along $\gamma_{+}$ between $0$
and $2d$. Observe that $B\left(\ell\right)$ is also in $P_{+}$.

Now we explain the choice between the two surfaces $P_{-}$ and $P_{+}$. Define
$p_{-}$, $q_{-}$ and $\gamma_{-}$ analogously. Suppose that
$\ell=\left[\gamma\right]$ with $\gamma\left(t_{j}\right)=q_{j}$ ($j=\pm$) and
let $W_{j}$ be the parallel vector field along $\gamma$ with
$W_{j}\left(t_{j}\right)=\gamma_{j}^{\prime}\left(0\right)$. Then choose the
sign $j$ in such a manner that $\left\\{W_{+},W_{-},\gamma^{\prime}\right\\}$
is a positively oriented frame along $\gamma$.

Figure 3: The outer billiard map on $\mathcal{L}_{\kappa}$ associated with $S$

Notice that in the hyperbolic case (in contrast with the Euclidean), parallel
translating a line $\ell$ a distance $d$ along unit speed geodesic rays
$\gamma_{1}$ and $\gamma_{2}$ orthogonal to $\ell$ depends on the initial
points $\gamma_{1}\left(0\right)$ and $\gamma_{2}\left(0\right)$ in
$\ell=\left[\gamma\right]$, even if $\gamma_{2}^{\prime}\left(0\right)$ is the
parallel transport of $\gamma_{1}^{\prime}\left(0\right)$ and along $\gamma$.

Given a strictly convex compact surface $S$ in the three sphere one can define
the outer billiard map on $\mathcal{C}$ associated with $S$ in a similar way
as for the Euclidean and hyperbolic cases. Only the domain of definition must
be further reduced. We call

$\mathcal{C}_{\operatorname{cut}\left(S\right)}=\left\\{\text{oriented great
circles intersecting }S\right\\}$ (1)

and $\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$ the Gauss image of the
surface $S$, that is,

$\mathcal{C}_{\operatorname{Gauss}\left(S\right)}=\left\\{\text{oriented great
circles contained in the tangent spaces to }S\right\\}$ (2)

(recall that the Gauss map of $S$, taking values in the Grassmannian of
oriented subspaces of dimension two of $\mathbb{R}^{4}$, assigns the subspace
$T_{p}S\subset\mathbb{R}^{4}$ to each point $p\in S$).

###### Proposition 1

Let $S$ be a strictly convex compact surface in $S^{3}$. The analogue of the
outer billiard map associated with $S$ above is well defined on
$\left(\mathcal{C}-\mathcal{C}_{\operatorname{Gauss}\left(S\right)}\right)-\mathcal{C}_{\operatorname{cut}\left(S\right)}$
and is a bijection onto this set.

As a first step towards the study of the billiards above, it would be
convenient to put them in a more general setting: We present a general
definition of outer billiard map which includes all the billiard maps we know
where the boundary of the billiard table is smooth.

###### Definition 2

Let $L$ be a smooth manifold and let $U$ be a connected open set in $L$ whose
boundary is an embedded hypersurface $M$ of $L$.

Let $T\in\left(0,\infty\right]$ and let $F:M\times\left(-T,T\right)\rightarrow
L$ be a smooth function such that:

a _)_ For each $p\in M$, the curve $\alpha_{p}:\left(-T,T\right)\rightarrow L$
defined by $\alpha_{p}(t)=F(p,t)$ satisfies that $\alpha_{p}\left(0\right)=p$
and $\alpha_{p}^{\prime}\left(0\right)\in T_{p}M$.

b _)_ The restrictions $F_{+}$, $F_{-}$ of $F$ to $M\times\left(0,T\right)$
and $M\times\left(-T,0\right)$, respectively, are both diffeomorphisms onto
$U.$

Then we say that the triple $\left(F,U,M\right)$ determines the outer billiard
map $B:U\rightarrow U$ defined by $B(q)=F_{+}(p,t)$, provided that
$F_{-}(p,-t)=q$ _(_ $t>0$_)_.

Although for most examples $L$ is actually a (pseudo-)Riemannian manifold and
the curves $\alpha_{p}$ are geodesics, we prefer to give the above more
general definition of outer billiard, since for instance in the main example
of this note it is more direct to give the tangent curves (which are defined
naturally) rather than their initial velocities, with the additional advantage
that we do not need to consider the (pseudo-)Riemannian structure on $L$ from
the beginning.

Examples. 1) The simplest outer billiard map, that is, the one induced by a
smooth closed embedded plane curve $\gamma$ with nowhere vanishing curvature,
satisfies the above definition with $L=\mathbb{R}^{2}$, $M$ the image of
$\gamma$ and $U$ its exterior, $T=\infty$ and
$\alpha_{\gamma\left(s\right)}\left(t\right)=\gamma\left(s\right)+t\gamma^{\prime}\left(s\right)$
for all $s,t$.

Higher dimensional outer billiards as in [22] also can be set in this
framework, see equation (3) below.

2) Theorem 3 below states that the outer billiard map on $\mathcal{L}_{0}$,
$\mathcal{L}_{-1}$ or $\mathcal{C}$ associated with a strictly convex compact
surface $S$ presented above satisfies the conditions of Definition 2, provided
that $S$ is quadratically convex, that is, the shape operator at any point of
$S$ is definite. For instance, for the outer billiard map on $\mathcal{C}$
associated with a surface $S$, the manifold $L$ is
$\mathcal{C}-\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$,
$U=L-\mathcal{C}_{\operatorname{cut}\left(S\right)}$ and $T=\pi/2$.

3) Following Tabachnikov’s article [22, page 235], the definition covers also
the case of inner billiards for a compact quadratically convex hypersurface
$S$ in $\mathbb{R}^{m}$: Let $n:S\rightarrow S^{m-1}$ denote the inward
pointing unit normal vector field. Let

$\mathcal{L}_{\operatorname{normal}\left(S\right)}=\left\\{\left[t\mapsto
p+\delta tn\left(p\right)\right]\mid p\in S\text{,\ }\delta=\pm 1\right\\}$

and $L=\mathcal{L}_{0}-\mathcal{L}_{\operatorname{normal}\left(S\right)}$. Let
$U=\left\\{\ell\in L\mid\ell\text{ intersects the interior of }S\right\\}$,
whose boundary in $L$ is the set $M$ of oriented lines tangent to $S$, which
may be identified with $T^{1}S$. For $v\in T_{p}^{1}S$ define the curve
$c_{v}:\left(-\pi/2,\pi/2\right)\rightarrow T^{1}\mathbb{R}^{m}$ by

$c_{v}\left(t\right)=\left(\cos t\right)v+\left(\sin
t\right)n\left(p\right)\text{.}$

Let $F:T^{1}S\times\left(-\pi/2,\pi/2\right)\rightarrow L$ be given by
$F\left(v,t\right)=\left[\gamma_{c_{v}\left(t\right)}\right]$. Then the triple
$\left(F,U,T^{1}S\right)$ determines an outer billiard map which coincides
with the inner billiard map with table the interior of $S$ (except on
$\mathcal{L}_{\operatorname{normal}\left(S\right)}$, on which the latter acts
simply by changing the orientation). Observe that
$\left(\alpha_{v}\right)^{\prime}\left(0\right)\in
T_{\left[\gamma_{v}\right]}M$ by Lemma 14 below, since the corresponding
Jacobi field along $\gamma_{v}$ vanishes at $s=0$.

We comment that by means of the free $\mathbb{Z}_{2}$-action $\ast$ on
$T^{1}S\times\left(0,\pi\right)$ defined by
$\varepsilon\ast\left(v,t\right)=\left(-v,\pi-t\right)$ ($\varepsilon=\pm 1$)
and a similar one on $T^{1}S\times\left(-\pi,0\right)$ one can make the inner
billiard almost fit into Definition 2, without having to omit the normal lines
to $S$.

The next theorem states that, under the stronger condition that $S$ is
quadratically convex, the outer billiard maps on $\mathcal{G}_{\kappa}$
associated with $S$ may also be written in the language of Definition 2. This
encompasses the result that they are smooth (and in particular,
diffeomorphisms).

On a strictly convex surface $S$ in $M_{\kappa}$ we consider the complex
structure $i$ defined by $i_{p}\left(u\right)=n\left(p\right)\times u$ for
$p\in S$, where $n:S\rightarrow TM_{\kappa}$ is the unit normal vector field
pointing inwards and $\times$ is the cross product associated with the
canonical orientation of $M_{\kappa}$. By abuse of notation we will write $iu$
omitting the foot point.

Given $v\in TM_{\kappa}$, we denote by $\gamma_{v}$ the unique geodesic in
$M_{\kappa}$ with initial velocity $v$.

###### Theorem 3

Suppose that $\kappa=0$ or $-1$, that is, $M_{\kappa}$ is Euclidean or
hyperbolic space, and let $S$ be a quadratically convex compact surface of
$M_{\kappa}$. Let

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\left\\{\text{oriented lines tangent to }S\right\\}\cong T^{1}S$
$\displaystyle\text{and\ \ \ \ \ \ \ \ }\mathcal{U}$ $\displaystyle=$
$\displaystyle\left\\{\text{oriented lines which do not intersect
}S\right\\}\text{.}$

For each $u\in T^{1}S$ and $t\in\mathbb{R}$, let $u_{t}\in T^{1}M_{\kappa}$ be
the parallel transport of $u$ between $0$ and $t$ along $\gamma_{iu}$. Let

$F:\mathcal{M}\times\mathbb{R}\cong
T^{1}S\times\mathbb{R}\rightarrow\mathcal{L}_{\kappa}\text{,\ \ \ \ \ \ \
}F(u,t)=\left[\gamma_{u_{t}}\right].$

Then $(F,\mathcal{U}$,$\mathcal{M})$ determines an outer billiard map that
coincides with the outer billiard map on $\mathcal{L}_{\kappa}$ associated
with $S$.

The analogous statement for the outer billiard map on $\mathcal{C}$ associated
with a surface is true.

The choice of $\mathcal{M}$ as the boundary of the billiard table was inspired
by [7].

In [22] (see also [5, 23]), Tabachnikov generalized planar outer billiards
(Figure 1) to even-dimensional standard symplectic space
$(\mathbb{R}^{2n},\omega)$ as follows: given a smooth quadratically convex
compact hypersurface $M$ in $\mathbb{R}^{2n}$, the restriction of $\omega$ to
each tangent space $T_{q}M$ has a $1$-dimensional kernel, called the
characteristic line. Tabachnikov showed that for each point $p$ outside of
$M$, exactly two characteristic lines of $M$ pass through $p$, so that a
choice of orientation induces a well-defined outer billiard map. Moreover, he
proved that the map is symplectic.

In the phrasing of Definition 2, the billiard above is the one determined by
$\left(F,U,M\right)$, where $U$ is the exterior of $M$ in
$\mathbb{R}^{2n}\cong\mathbb{C}^{n}$, and

$F:M\times\mathbb{R}\rightarrow\mathbb{C}^{n}\text{,\ \ \ \ \ \ \
}F\left(q,t\right)=q+tin\left(q\right)\text{,}$ (3)

with $n$ the outward pointing unit normal vector field of $M$.

We will show that our outer billiard can also be defined in a similar manner.
For that, we recall the canonical Kähler or Poisson structures on
$\mathcal{G}_{\kappa}$. See the details in Subsection 2.2.

For $\kappa=1,0,-1$, given $\ell\in\mathcal{G}_{\kappa}$, the rotation in
$M_{\kappa}$ through an angle of $\pi/2$ which fixes $\ell$ induces a
diffeomorphism of $\mathcal{G}_{\kappa}$ which fixes $\ell$, whose
differential at $\ell$ is a linear transformation of
$T_{\ell}\mathcal{G}_{\kappa}$ squaring to $-$id. This yields a complex
structure $\mathcal{J}$ on $\mathcal{G}_{\kappa}$.

For $\kappa=1,-1$, the manifold $\mathcal{G}_{\kappa}$ has two canonical
pseudo-Riemannian metrics $g_{\times}$ and $g_{K}$, induced by the cross
product on $M_{\kappa}$ and the Killing form on Iso$\left(M_{\kappa}\right)$,
respectively. The metric $g_{\times}$ is also well defined on
$\mathcal{G}_{0}$. Moreover, in any case, $\left(g_{K},\mathcal{J}\right)$ and
$\left(g_{\times},\mathcal{J}\right)$ are Kähler structures on
$\mathcal{G}_{\kappa}$. In the Euclidean case, the Killing form $g_{K}$
degenerates, however, on $\mathcal{G}_{0}$ there is a weaker structure also
compatible with $\mathcal{J}$, namely, a Poisson bivector field $\mathcal{P}$.

The next two theorems assert that for $\kappa=\pm 1$, the outer billiard on
$\mathcal{G}_{\kappa}$ is obtained by an analogue of Tabachnikov’s
construction, provided $\mathcal{G}_{\kappa}$ is endowed with the Kähler
structure $\left(g_{K},\mathcal{J}\right)$, and moreover it is a
symplectomorphism with respect to the associated fundamental form. Afterwards
we show that this is not the case for $\left(g_{\times},\mathcal{J}\right)$
and that $\mathcal{P}$ is preserved by the outer billiard map on
$\mathcal{G}_{0}$.

###### Theorem 4

For $\kappa=\pm 1$, let $S$ be a quadratically convex compact surface in
$M_{\kappa}$. Let $\mathcal{G}_{\kappa}$ be endowed with the Kähler structure
$\left(g_{K},\mathcal{J}\right)$. Then the outer billiard on
$\mathcal{G}_{\kappa}$ determined by $\left(F,\mathcal{U},\mathcal{M}\right)$
satisfies

$F\left(u,t\right)=\Gamma_{\mathcal{J}\left(N\left(u\right)\right)}\left(t\right),$

for all $u\in T^{1}S\cong\mathcal{M}$ and all $t$, where $N$ is the unit
normal vector field on $\mathcal{M}$ pointing outwards and $\Gamma_{\xi}$
denotes the geodesic in $(\mathcal{G}_{\kappa},g_{K})$ with initial velocity
$\xi$.

###### Theorem 5

For $\kappa=\pm 1$, the outer billiard map associated with a quadratically
convex compact surface in $M_{\kappa}$ is a symplectomorphism with respect to
the fundamental symplectic form of
$\left(\mathcal{G}_{\kappa},g_{K},\mathcal{J}\right)$.

###### Proposition 6

For $\kappa=0,1,-1$, the billiard map does not preserve the fundamental
symplectic form of the Kähler structure $\left(g_{\times},\mathcal{J}\right)$
on $\mathcal{G}_{\kappa}$.

In the Euclidean case, the outer billiard map associated with a quadratically
convex compact surface $S$ preserves parallelism, yielding an $S^{2}$-worth of
planar outer billiards as in Figure 1: The complements of the shadows of $S$
provide billiard tables with shapes changing smoothly (for each $v$ in
$\mathbb{R}^{3}$, the orthogonal projection of $S$ onto any plane orthogonal
to $v$ determines a closed convex curve in the plane called the shadow of $S$
along $v$). For each fixed direction the outer billiard map on the plane is
well-understood, in particular, it preserves the area. For the more general
system on $\mathcal{G}_{0}$ this translates into the assertion in the
following proposition.

###### Proposition 7

The outer billiard map associated with a quadratically convex compact surface
in $\mathbb{R}^{3}$ preserves the canonical Poisson structure $\mathcal{P}$ on
$\mathcal{G}_{0}$, and its symplectic leaves, which are the submanifolds of
parallel lines. In particular, the restrictions to them are symplectomorphims.

Regarding the dynamics of the outer billiard map on $\mathcal{L}_{-1}$
associated with a surface in $H^{3}$, most of the natural questions remain
unanswered. In Propositions 8 and 9 below we present the first steps into this
subject, that hint at its complexity. The hyperbolic case is more interesting
than the Euclidean one, because the lines $\ell$, $B\left(\ell\right)$,
$B^{2}\left(\ell\right),\dots$ in $H^{3}$ are in general not parallel, in the
sense that they are not orthogonal to a fixed totally geodesic surface, as in
the Euclidean case. This comes from the fact that if you parallel translate a
line $\ell$ in $H^{3}$ along a line $\ell^{\prime}$ orthogonal to it, then
$\ell^{\prime}$ is the unique line preserved by the motion, in contrast with
the Euclidean case. The following proposition illustrates this phenomenon.

###### Proposition 8

Let $0<\theta<\pi/2$. There exist a quadratically convex compact surface $S$
in $H^{3}$ and an oriented line $\ell$ not intersecting $S$ such that
$B^{3}\left(\ell\right)$ and $\ell$ intersect at a point forming the angle
$\theta$, where $B$ is the outer billiard map on $\mathcal{L}_{-1}$ associated
with $S$.

Similar arguments show the existence of $\ell$ as above such that
$B^{3}\left(\ell\right)$ is different from $\ell$ and asymptotic to it.

For $\kappa=0,-1$, let
$\varpi:P=T^{1}M_{\kappa}\rightarrow\mathcal{L}_{\kappa}$ be the tautological
line bundle, which is an $\left(\mathbb{R},+\right)$-principal bundle. The
right action $\rho:P\times\mathbb{R}\rightarrow P$ is given by
$\rho\left(u,t\right)=\gamma_{u}^{\prime}\left(t\right)$.

Let $S$ be a strictly convex compact surface in $M_{\kappa}$ and let
$\mathcal{U}$ as before the set of oriented lines in $M_{\kappa}$ which do not
intersect $S$.

The line bundle
$P_{\mathcal{U}}=\varpi^{-1}\left(\mathcal{U}\right)\rightarrow\mathcal{U}$
has two distinguished sections $\sigma_{+}$ and $\sigma_{-}$: For $j=\pm$,
$\sigma_{j}\left(\left[\gamma\right]\right)=\gamma^{\prime}\left(0\right)$ if
$\gamma\left(0\right)=q_{j}\left(\ell\right)$, where $q_{+}$ and $q_{-}$ are
as in the definition of the outer billiard map on $\mathcal{L}_{\kappa}$
associated with $S$ in the introduction. They allow us to define the notion of
holonomy of a periodic point of the corresponding billiard map.

Let $\ell_{0}\in\mathcal{U}$ be a periodic point of $B$, say
$B^{n}\left(\ell_{0}\right)=\ell_{0}$ for some $n>1$ (in particular, $n\geq
3$) and suppose that $B^{m}\left(\ell_{0}\right)\neq\ell_{0}$ for $0<m<n$. For
$0\leq k\leq n$ call $\ell_{k}=B^{k}\left(\ell_{0}\right)$ and let $d_{k}$ be
the signed distance between $q_{+}\left(\ell_{k}\right)$ and
$q_{-}\left(\ell_{k}\right)$; more precisely, if
$\ell_{k}=\left[\gamma_{k}\right]$ with
$\gamma_{k}\left(0\right)=q_{-}\left(\ell_{k}\right)$, then $d_{k}$ is defined
by $\gamma_{k}\left(d_{k}\right)=q_{+}\left(\ell_{k}\right)$. Then the
holonomy of $B$ at $\ell$ is the number $d=\sum_{k=0}^{n-1}d_{k}$. We use the
word holonomy in a wider sense: It makes sense only for periodic points and it
is not associated with a particular connection, but rather with a combination
of the flat connections induced by $\sigma_{\pm}$ and the Levi-Civita
connection on $M_{\kappa}$ along the shortest segments joining $\ell_{k}$ with
$\ell_{k+1}$.

It is easy to see that the holonomy of a periodic point of the billiard map is
zero in the Euclidean case. We will give an example of a surface in hyperbolic
space whose associated billiard map has a periodic point with nontrivial
holonomy.

###### Proposition 9

There exist a quadratically convex compact surface $S$ in $H^{3}$ and an
oriented line $\ell$ not intersecting $S$ which is a periodic point of the
associated outer billiard map and whose holonomy is not zero.

Apart from the usual questions on the dynamics of a general system, the
following is specific of the outer billiard map $B$ on $\mathcal{L}_{-1}$
associated with a surface $S$ in $H^{3}$: For $S$ the round sphere in
hyperbolic space, $B$ preserves parallelism (in the sense that for any
$\ell\in\mathcal{L}_{-1}$, $B^{n}\left(\ell\right)$ remains orthogonal to a
certain totally geodesic surface). We wonder whether this property
characterizes the sphere. This is related to the issue of defining shadows of
convex surfaces in $H^{3}$.

##  2 Preliminaries

###  2.1 The Jacobi fields of the three dimensional space forms

We recall that $(M_{\kappa},\left\langle,\right\rangle_{\kappa})$ denotes the
three-dimensional complete simply connected manifold of constant sectional
curvature $\kappa=0,1,-1$, that is, $\mathbb{R}^{3}$, $S^{3}$ and hyperbolic
space $H^{3}$. The curvature tensor of $M_{\kappa}$ is given by

$R_{\kappa}\left(x,y\right)z=\kappa\left(\left\langle
z,x\right\rangle_{\kappa}y-\left\langle z,y\right\rangle_{\kappa}x\right),$
(4)

for vector fields $x,y,z$ on $M_{\kappa}$.

Let $M$ be a complete Riemannian manifold and let $\gamma$ be a unit speed
geodesic of $M$. A Jacobi field $J$ along $\gamma$ is by definition a vector
field along $\gamma$ arising via a variation of geodesics as follows: Let
$\delta>0$ and $\phi:\mathbb{R}\times\left(-\delta,\delta\right)\rightarrow M$
be a smooth map such that $r\mapsto\phi\left(r,s\right)$ is a geodesic for
each $s\in\left(-\delta,\delta\right)$ and
$\phi\left(r,0\right)=\gamma\left(r\right)$ for all $r$. Then

$J\left(r\right)=\left.\tfrac{d}{ds}\right|_{0}\phi\left(r,s\right)\text{.}$

It is well known that Jacobi fields along a geodesic $\gamma$ and orthogonal
to $\gamma^{\prime}$ of $M_{\kappa}$ are exactly the vector fields $J$ along
$\gamma$ with $\left\langle J,\gamma^{\prime}\right\rangle_{\kappa}=0$
satisfying the equation

$\tfrac{D^{2}J}{dr^{2}}+\kappa J=0.$ (5)

In the rest of the article, following a common abuse of notation, given a
smooth vector field $J$ along a curve $\gamma$, we write
$J^{\prime}=\frac{DJ}{dr}$ if there is no danger of confusion.

Recall that a Jacobi field $J$ along $\gamma$ is determined by the values
$J\left(0\right)$ and $J^{\prime}\left(0\right)$. Suppose that a Jacobi field
$J$ along $\gamma$ satisfies
$J\left(0\right)=u+a\gamma^{\prime}\left(0\right)$ and
$J^{\prime}\left(0\right)=v+b\gamma^{\prime}\left(0\right)$ where
$a,b\in\mathbb{R}$ and $u,v\in\gamma^{\prime}\left(0\right)^{\bot}$. Let $U$
and $V$ be the parallel vector fields along $\gamma$ with $U\left(0\right)=u$
and $V\left(0\right)=v$. Then

$J\left(r\right)=c_{\kappa}\left(r\right)~{}U(r)+s_{\kappa}\left(r\right)~{}V(r)+\left(a+rb\right)\gamma^{\prime}(r)\text{,}$
(6)

where

$\begin{array}[]{ccc}c_{1}(r)=\cos
r\text{,}&c_{0}(r)=1\text{,}&c_{-1}(r)=\cosh r\text{,}\\\ s_{1}(r)=\sin
r\text{,}&s_{0}(r)=r\text{,}&s_{-1}(r)=\sinh r\end{array}$

(in particular, $s_{\kappa}^{\prime}=c_{\kappa}$ and
$c_{\kappa}^{\prime}=-\kappa s_{\kappa}$). This expression will allow us to
perform most computations without having to resort to coordinates of
$M_{\kappa}$ or a particular model of it.

For $\kappa=0,1,-1$, we presented in the introduction the space
$\mathcal{G}_{\kappa}$ of all the oriented geodesics of $M_{\kappa}$. Next we
see that tangent vectors to $\mathcal{G}_{\kappa}$ at an oriented geodesic
$\left[\gamma\right]$ may be identified with Jacobi fields along $\gamma$.

Let $\gamma$ be a complete unit speed geodesic of $M_{\kappa}$ and let
$\mathfrak{J}_{\gamma}$ be the space of all Jacobi fields along $\gamma$ which
are orthogonal to $\gamma^{\prime}$. There is a canonical isomorphism

$T_{\gamma}:\mathfrak{J}_{\gamma}\rightarrow
T_{[\gamma]}\mathcal{\mathcal{G}_{\kappa}}\text{,}\hskip
28.45274ptT_{\gamma}(J)=\left.{\tfrac{d}{ds}}\right|_{0}[\gamma_{s}]\text{,}$
(7)

where $\gamma_{s}$ is any variation of $\gamma$ by unit speed geodesics
associated with $J$. Moreover, if $J$ is the Jacobi field associated to a
variation $\phi:\mathbb{R}\times\left(-\delta,\delta\right)\rightarrow
M_{\kappa}$ of $\gamma$ by unit speed geodesics ($J$ is not necessarily
orthogonal to $\gamma^{\prime}$), then

$T_{\gamma}(J^{N})=\left.\tfrac{d}{ds}\right|_{0}[\phi_{s}],$ (8)

where $J^{N}\left(r\right)=J\left(r\right)-\left\langle
J\left(r\right),\gamma^{\prime}\left(r\right)\right\rangle_{\kappa}\gamma^{\prime}\left(r\right)$
(see Section 2 in [13] and [18]).

###  2.2 Kähler structures on the manifolds of oriented geodesics

In the introduction we mentioned the two canonical Kähler structures
$\left(g_{K},\mathcal{J}\right)$ and $\left(g_{\times},\mathcal{J}\right)$ on
the space of oriented geodesics $\mathcal{G}_{\kappa}$ for $\kappa=1,-1$, and
the Kähler structure $\left(g_{K},\mathcal{J}\right)$ and the Poisson bivector
field $\mathcal{P}$ on $\mathcal{G}_{0}$. Next we present the precise
definitions, in terms of the isomorphism (7). We also include the expressions
of the associated fundamental forms (see [12, 17, 18, 6, 1, 9] (Theorem 1 in
[18] is valid also for $\kappa=0,1$). The geometry of these spaces of
geodesics has been studied for instance in [10, 11, 19]. See for instance in
[8] or [14] the proof that $\mathcal{C}$ is diffeomorphic to $S^{2}\times
S^{2}$.

Given $\ell=\left[\gamma\right]\in\mathcal{G}_{\kappa}$, the linear complex
structure $\mathcal{J}_{\ell}$ on $\mathfrak{J}_{\gamma}\cong
T_{\ell}\mathcal{G}_{\kappa}$ is defined by

$\mathcal{J}_{\ell}\left(J\right)=\gamma^{\prime}\times J,\hskip
14.22636pt\text{for all}\hskip 7.11317ptJ\in\mathfrak{J}_{\gamma}\cong
T_{\ell}\mathcal{G}_{\kappa}\text{.}$ (9)

The square norms of the metrics $g_{K}$ and $g_{\times}$ may be written as
follows (we put $\left\|Z\right\|=\left\langle Z,Z\right\rangle$): For
$J\in\mathfrak{J}_{\gamma}\cong T_{[\gamma]}\mathcal{\mathcal{G}_{\kappa}}$,

$\|J\|_{\times}=\langle\gamma^{\prime}\times
J,J^{\prime}\rangle_{\kappa}\text{\ \ \ \ \ \ and\ \ \ \ \ \
}\|J\|_{K}=\left|J\right|_{\kappa}^{2}+\kappa\left|J^{\prime}\right|_{\kappa}^{2}\text{.}$

Notice that by (5) the right hand sides are constant functions, so the left
hand sides are well defined. Proposition 2 in [18] provides a geometric
interpretation for the metrics $g_{\times}$ and $g_{K}$ in the case
$\kappa=-1$: whether a curve in $\mathcal{G}_{-1}$ is space-like or time-like
is related to the dichotomies positive versus negative screw for the former
and translation versus rotation for the latter. By polarization we have

$\displaystyle 2\,g_{\times}\left(I,J\right)$ $\displaystyle=$
$\displaystyle\langle I\times J^{\prime}+J\times
I^{\prime},\gamma^{\prime}\rangle_{\kappa}\hskip 14.22636pt\text{for}\
\kappa=-1,0,1\text{;}$ (10) $\displaystyle g_{K}\left(I,J\right)$
$\displaystyle=$ $\displaystyle\langle I,J\rangle_{\kappa}+\kappa\langle
I^{\prime},J^{\prime}\rangle_{\kappa}\hskip 14.22636pt\text{for}\ \kappa=\pm
1\text{.}$ (11)

The second one is the push down onto $\mathcal{G}_{\kappa}$ from the left
invariant pseudo-Riemannian metric on the Lie group
Iso${}_{0}\left(M_{\kappa}\right)$ given at the identity by the Killing form
of its Lie algebra. It is Riemannian for $\kappa=1$ and split for $\kappa=-1$.
It is well known that $(\mathcal{G}_{\kappa},g_{\times},\mathcal{J})$ is
Kähler for $\kappa=0,1,-1$ (see, for example, page 753 of [9]).

The associated fundamental forms are given by

$\displaystyle\omega_{\times}\left(I,J\right)$ $\displaystyle=$ $\displaystyle
g_{\times}\left(\mathcal{J}\left(I\right),J\right)=\tfrac{1}{2}(\langle
I^{\prime},J\rangle_{\kappa}-\langle I,J^{\prime}\rangle_{\kappa})\text{,}$
(12) $\displaystyle\omega_{K}\left(I,J\right)$ $\displaystyle=$ $\displaystyle
g_{K}\left(\mathcal{J}\left(I\right),J\right)=\langle I\times J+\kappa
I^{\prime}\times J^{\prime},\gamma^{\prime}\rangle_{\kappa}\text{.}$ (13)

The bilinear form $g_{K}$ degenerates for $\kappa=0$. On $\mathcal{G}_{0}$ we
have the canonical Poisson structure, given by

$\mathcal{P}\left(\ell\right)=J_{\ell}\wedge\mathcal{J}_{\ell}\left(J\right)$

where $J$ is any parallel Jacobi field along $\ell$ with
$\left\|J\right\|\equiv 1$. Although no such a section $\ell\mapsto
J_{\ell}\in T_{\ell}\mathcal{G}_{0}$ exists globally (otherwise, it would
induce a unit vector field on the 2-sphere), $\mathcal{P}$ is easily seen to
be well defined and smooth; the Schouten bracket
$\left[\mathcal{P},\mathcal{P}\right]$ vanishes, since the distribution on
$\mathcal{G}_{0}$ induced by $\mathcal{P}$ is integrable. In fact, the
symplectic leaves are the submanifolds of parallel lines. The proof of
Proposition 7 is straighforward from the well-known fact that the planar outer
billiard map preserves the area and we omit it.

##  3 Proofs of the results

Before proving Proposition 1 we comment on the set
$\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$ defined in (2). We define
the map $c:S\rightarrow\mathcal{C}$ as follows: Given $p\in S$, let
$c\left(p\right)$ be the oriented great circle obtained by intersecting
$S^{3}$ with the subspace $T_{p}S\subset\mathbb{R}^{4}$, endowed with the
orientation induced by that of $T_{p}S$. Equivalently, and without using the
immersion of the sphere in $\mathbb{R}^{4}$, for any positively oriented
orthonormal basis $\left\\{u,v\right\\}$ of $T_{p}S$,
$c\left(p\right)=\left[C_{p}\right]$, where

$C_{p}\left(s\right)=\text{Exp}_{p}\left(\tfrac{\pi}{2}\left(\cos s~{}u+\sin
s~{}v\right)\right)\cong\cos s~{}u+\sin s~{}v\text{.}$

With this notation, $\mathcal{C}_{\text{Gauss}\left(S\right)}$ is the image of
$c$. Notice that the fact that $S$ is compact and strictly convex implies that
if an oriented plane is in $\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$,
then the same plane with the opposite orientation is also in
$\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$.

Proof of Proposition 1. We verify that the same procedure as for the Euclidean
and hyperbolic spaces applies here. By [4], $S$ is contained in a hemisphere,
say $S_{+}^{3}=\left\\{x\in S^{3}\mid x_{0}>0\right\\}$, where
$x_{0},\dots,x_{3}$ are the usual coordinates in $\mathbb{R}^{4}$. Consider
the central projection $\Pi$ from
$\left\\{1\right\\}\times\mathbb{R}^{3}\cong\mathbb{R}^{3}$ to $S_{+}^{3}$
(called the Beltrami map), which preserves both strict and quadratical
convexity, since planes correspond with half great spheres and the order of
contact is preserved by diffeomorphisms.

Let $c\in\mathcal{C}-\mathcal{C}_{\operatorname{cut}\left(S\right)}$. If $c$
is not contained in the equator $x_{0}=0$, then $\Pi^{-1}\left(c\right)$ is an
oriented line $\ell$ in $\mathbb{R}^{3}$. Exactly two planes in
$\mathbb{R}^{3}$ containing $\ell$ will intersect $\Pi^{-1}\left(S\right)$ at,
say, the tangent points $\bar{p}_{1}$ and $\bar{p}_{2}$. They project to two
half great spheres containing $c$, which are tangent to $S$ at
$p_{j}=\Pi\left(\bar{p}_{j}\right)$, $j=1,2$. These points play the roles of
the corresponding points in the definition of the outer billiard maps on
$\mathcal{L}_{0}$ and $\mathcal{L}_{-1}$ in the introduction. If $c$ is in the
equator $x_{0}=0$ and $u\in e_{0}^{\bot}$ is a unit vector orthogonal to $c$,
then the planes orthogonal to $u$ project to half great spheres containing
$c$. Among these parallel planes, exactly two of them intersect
$\Pi^{-1}\left(S\right)$ and one obtains as before two great spheres
containing $c$, which are tangent to $S$ at some points $p_{1}$ and $p_{2}$.

Now, let us suppose additionally that
$c\notin\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$. In the following
argument the Beltrami map is not needed any more. Let $d$ be the distance
between $p_{1}$ and $c$. If $d<\pi/2$, there exists exactly one point $q_{1}$
on $c$ realizing the distance. If $d=\pi/2$, since $c$ is contained in the
great sphere Exp${}_{p_{1}}\left(T_{p_{1}}S\right)$, which is tangent to
$p_{1}$, then $c\in\mathcal{C}_{\operatorname{Gauss}\left(S\right)}$, which is
a contradiction. The proof continues as in the Euclidean and hyperbolic cases.

Next we prove some technical lemmas, working towards the proof of Theorem 3.

Before we state the next lemma we give some geometric intuition for it.
Consider the situation in which $S$ is a strictly convex compact surface in
$M_{\kappa}$, $\mathcal{U}$ is the collection of oriented geodesics not
intersecting $S$, and its boundary $\mathcal{M}$ is the collection of
geodesics tangent to $S$. Given $\ell\in\mathcal{M}$, one may consider three
perturbations of $\ell$ which stay in $\mathcal{M}$: one which skates along
$S$ in the direction of $\ell$, one which parallel translates $\ell$ along the
surface in the direction orthogonal to $\ell$, and one which rotates $\ell$,
maintaining the point of tangency.

###### Lemma 10

Let $S$ be a strictly convex compact surface in $M_{\kappa}$, let $u\in
T_{p}^{1}S$ and call $v=iu$. For $m=1,2,3,$ let $w_{m}:\mathbb{R}\rightarrow
T^{1}S$ be the curve defined by

$w_{1}(s)=\sigma_{u}^{\prime}(s)\text{,\ \ \ \ \ \ \ \
}w_{2}(s)=P_{0,s}^{\sigma_{v}}(u)\hskip 21.33955pt\text{and}\hskip
21.33955ptw_{3}(s)=\cos s~{}u+\sin s~{}v\text{,}$

where $\sigma_{v}$ is the geodesic on $S$ with initial velocity $v$ and
$P_{0,s}^{\sigma}$ denotes the parallel transport on $S$ along $\sigma$
between $0$ and $s$. Then
$\mathcal{\tilde{B}}=\left\\{w_{1}^{\prime}\left(0\right),w_{2}^{\prime}\left(0\right),w_{3}^{\prime}\left(0\right)\right\\}$
is a basis of $T_{u}T^{1}S$.

Proof. Let $\pi:TS\rightarrow S$ be the canonical projection and let
$\mathcal{K}_{u}:T_{u}TS\rightarrow T_{p}S$ be the connection operator. We
claim that under the linear isomorphism

$\varphi_{u}:T_{u}TS\rightarrow T_{p}S\times T_{p}S\text{,\ \ \ \ \ \ \ \ \ \
\ \ }\varphi_{u}(\xi)=(d\pi_{u}\xi,\mathcal{K}_{u}\xi)\text{,}$

(see for instance [2]) the elements of $\mathcal{\tilde{B}}$ are mapped to
$(u,0)$, $(v,0)$ and $(0,v)$, respectively, which are linearly independent. We
compute

$d\pi_{u}(w_{1}^{\prime}(0))=\left.\tfrac{d}{ds}\right|_{0}\pi(w_{1}(s))=\left.\tfrac{d}{ds}\right|_{0}\sigma_{u}(s)=u,$

and, by definition of $\mathcal{K}_{u}$,

$\mathcal{K}_{u}(w_{1}^{\prime}(0))=\left.\tfrac{D}{ds}\right|_{0}w_{1}(s)=\left.\tfrac{D}{ds}\right|_{0}\sigma_{u}^{\prime}(s)=0.$

Hence, $\varphi_{u}(w_{1}^{\prime}(0))=(u,0)$. The other cases are similar.

We recall some notation. We denote by $\gamma_{w}$ the unique geodesic in
$M_{\kappa}$ with initial velocity $w\in T^{1}M_{\kappa}$. Given a strictly
convex compact surface $S$ in $M_{\kappa}$, we call $n$ the unit normal vector
field on $S$ pointing inward. The complex structure $i$ on $S$ is defined by
$iw=n\left(p\right)\times w$, for $w\in T_{p}S$.

Given $p\in S$, the shape operator $A_{p}:T_{p}S\rightarrow T_{p}S$ is defined
by

$A_{p}\left(x\right)=-\nabla_{x}n\text{,}$ (14)

where $\nabla$ denotes the Levi-Civita connection of $M_{\kappa}$. We assume
that $A_{p}$ is positive definite at each $p\in S$ (that is, $S$ is
quadratically convex).

Let $u\in T_{p}^{1}S$ and call $v=iu\in T_{p}^{1}S$. We consider the matrix of
$A_{p}$ with respect to the orthonormal basis $\left\\{u,v\right\\}$ and call
$b_{ij}$ its entries, that is,
$\left[A_{p}\right]_{\left\\{u,v\right\\}}=\left(\begin{array}[]{cc}b_{11}&b_{12}\\\
b_{21}&b_{22}\end{array}\right)$. In particular, we have that $b_{12}=b_{21}$.

We call $u_{r}$, $v_{r}$ and $n_{r}$ the vectors in
$T_{\gamma_{v}\left(r\right)}M_{\kappa}$ obtained by parallel transporting
$u,v$ and $n\left(p\right),$ respectively, along $\gamma_{v}$, between $0$ and
$r$.

For $m=1,2,3$, let $w_{m}$ be the curve in $T^{1}S$ defined in Lemma 10 and
let $f_{m}:\mathbb{R}^{2}\rightarrow M_{\kappa}$ be the parametrized surface
in $M_{\kappa}$ defined by

$f_{m}\left(r,s\right)=\gamma_{iw_{m}(s)}\left(r\right).$

In particular $f_{m}\left(r,0\right)=\gamma_{v}\left(r\right)$. We call
$\sigma_{m}=\pi\circ w_{m}=f_{m}\left(0,\cdot\right)$; we have that
$\sigma_{1}=\sigma_{u}$, $\sigma_{2}=\sigma_{v}$ and $\sigma_{3}\equiv p$.

###### Lemma 11

For $m=1,2,3$, the Jacobi vector field $K_{m}$ along $\gamma_{v}$ associated
with the geodesic variation $f_{m}$, that is,

$K_{m}(r)=\left.\tfrac{d}{ds}\right|_{0}f_{m}\left(r,s\right)\text{,}$ (15)

is given by

$\begin{array}[]{l}K_{1}(r)=c_{\kappa}(r)~{}u_{r}+b_{21}s_{\kappa}(r)~{}n_{r},\\\
K_{2}(r)=v_{r}+b_{22}s_{\kappa}(r)~{}n_{r},\\\
K_{3}(r)=-s_{\kappa}(r)~{}u_{r}.\end{array}$ (16)

Proof. We compute the initial values of $K_{m}$ and $K_{m}^{\prime}$ and use
(6). We write down the details for $m=1$. The other cases are similar. We have
that

$K_{1}(0)=\left.\tfrac{d}{ds}\right|_{0}f_{1}\left(0,s\right)=\left.\tfrac{d}{ds}\right|_{0}\sigma_{1}(s)=u$

and

$K_{1}^{\prime}(0)=\left.\tfrac{D}{\partial
r}\right|_{0}\left.\tfrac{\partial}{\partial
s}\right|_{0}f_{1}\left(r,s\right)=\left.\tfrac{D}{\partial
s}\right|_{0}\left.\tfrac{\partial}{\partial
r}\right|_{0}f_{1}\left(r,s\right)=\left.\tfrac{D}{\partial
s}\right|_{0}iw_{1}(s)\text{.}$

We compute the coordinates of $K_{1}^{\prime}(0)$ with respect to the
orthonormal basis $\\{u,v,n(p)\\}$ of $T_{p}M_{\kappa}$. To obtain
$\left\langle K_{1}^{\prime}(0),n(p)\right\rangle_{\kappa}$, we observe that
$\left\langle iw_{1}(s),n(\sigma_{1}(s))\right\rangle_{\kappa}=0$ for all $s$.
Hence,

$\left\langle\left.\tfrac{D}{\partial
s}\right|_{0}iw_{1}(s),n(p)\right\rangle_{\kappa}=-\left\langle
iu,\left.\tfrac{D}{\partial
s}\right|_{0}n(\sigma_{1}(s))\right\rangle_{\kappa}=-\left\langle
v,\nabla_{u}n\right\rangle_{\kappa}=\left\langle
v,A_{p}(u)\right\rangle_{\kappa}=\,b_{21}\text{.}$

In the same way, $\left\langle
K_{1}^{\prime}(0),u\right\rangle_{\kappa}=0=\left\langle
K_{1}^{\prime}(0),v\right\rangle_{\kappa}$. Therefore,
$K_{1}^{\prime}(0)=b_{21}n(p)$. Notice that $K_{m}$ is not necessarily
orthogonal to $\gamma_{v}^{\prime}$.

###### Lemma 12

For $m=1,2,3$, let $N_{m}$ be the vector field along $f_{m}$ defined as
follows: $N_{m}\left(r,s\right)$ is the parallel transport of
$n\left(f_{m}\left(0,s\right)\right)=n\left(\sigma_{m}\left(s\right)\right)$
between $0$ and $r$ along the geodesic
$\gamma_{iw_{m}(s)}=f_{m}\left(\cdot,s\right)$. Let

$\zeta_{m}\left(r\right)=\left\langle\left.\tfrac{D}{ds}\right|_{0}N_{m}\left(r,s\right),u_{r}\right\rangle_{\kappa}.$

Then

$\zeta_{1}\equiv-b_{11}\text{,\ \ \ \ \ \ }\zeta_{2}\equiv-b_{12}\text{ \ \ \
\ \ \ \ and\ \ \ \ \ \ }\zeta_{3}\equiv 0\text{.}$

Proof. We compute

$\displaystyle\zeta_{m}^{\prime}\left(r\right)$ $\displaystyle=$
$\displaystyle\tfrac{d}{dr}\left\langle\left.\tfrac{D}{ds}\right|_{0}N_{m}\left(r,s\right),u_{r}\right\rangle_{\kappa}=\left\langle\tfrac{D}{dr}\left.\tfrac{D}{ds}\right|_{0}N_{m}\left(r,s\right),u_{r}\right\rangle_{\kappa}$
$\displaystyle=$
$\displaystyle\left\langle\left.\tfrac{D}{ds}\right|_{0}\tfrac{D}{dr}N_{m}\left(r,s\right)+R_{\kappa}\left(\tfrac{\partial
f_{m}}{\partial r}\left(r,0\right),\tfrac{\partial f_{m}}{\partial
s}\left(r,0\right)\right)N_{m}\left(r,0\right),u_{r}\right\rangle_{\kappa}$
$\displaystyle=$ $\displaystyle\left\langle
R_{\kappa}(v_{r},K_{m}(r))n_{r},u_{r}\right\rangle_{\kappa}=\kappa\left\langle\left\langle
n_{r},v_{r}\right\rangle_{\kappa}K_{m}(r)-\left\langle
n_{r},K_{m}(r)\right\rangle_{\kappa}v_{r},u_{r}\right\rangle_{\kappa}=0\text{,}$

by (4). Hence, $\zeta_{m}$ is constant, equal to

$\zeta_{m}\left(0\right)=\left\langle\left.\tfrac{D}{ds}\right|_{0}N_{m}\left(0,s\right),u_{0}\right\rangle_{\kappa}=\left\langle\nabla_{\sigma_{m}^{\prime}\left(0\right)}n,u\right\rangle_{\kappa}=-\left\langle
A_{p}\left(\sigma_{m}^{\prime}\left(0\right)\right),u\right\rangle_{\kappa}\text{.}$

Now, the assertions follow from the definition of $\sigma_{m}$ and the values
of the entries of the matrix $\left[A_{p}\right]_{\left\\{u,v\right\\}}$.

###### Lemma 13

For $m=1,2,3$, let $W_{m}$ be the vector field along $f_{m}$ defined
analogously as $N_{m}$, with $w_{m}\left(s\right)$ instead of
$n\left(\sigma_{m}\left(s\right)\right)$. Let $Y_{m}$ be the vector field
along $\gamma_{v}=f_{m}\left(\cdot,0\right)$ given by

$Y_{m}\left(r\right)=\left.\tfrac{D}{ds}\right|_{0}W_{m}\left(r,s\right)$

Then

$Y_{1}(r)=\kappa s_{\kappa}(r)~{}v_{r}+b_{11}~{}n_{r}\text{,\ \ \ \ \ \
}Y_{2}(r)=b_{12}~{}n_{r}\text{ \ \ \ \ \ and\ \ \ \ \ \
}Y_{3}(r)=c_{\kappa}(r)~{}v_{r}.$

Proof. Observe that $\left\langle W_{m},W_{m}\right\rangle_{\kappa}$,
$\left\langle W_{m},N_{m}\right\rangle_{\kappa}$ and $\left\langle
W_{m},\frac{\partial f_{m}}{\partial r}\right\rangle_{\kappa}$ are constant
functions of the second variable $s$. Hence we can compute the components
$Y_{m}\left(r\right)$ with respect to the basis
$\left\\{u_{r},v_{r},n_{r}\right\\}$ of
$T_{\gamma_{v}\left(r\right)}M_{\kappa}$ as follows:

$\left\langle
Y_{m}\left(r\right),u_{r}\right\rangle_{\kappa}=\left\langle\left.\tfrac{D}{ds}\right|_{0}W_{m}\left(r,s\right),W_{m}\left(r,0\right)\right\rangle_{\kappa}=0.$

Also

$\displaystyle\left\langle Y_{m}\left(r\right),v_{r}\right\rangle_{\kappa}$
$\displaystyle=$
$\displaystyle\left\langle\left.\tfrac{D}{ds}\right|_{0}W_{m}\left(r,s\right),v_{r}\right\rangle_{\kappa}=-\left\langle
W_{m}\left(r,0\right),\left.\tfrac{D}{ds}\right|_{0}\gamma_{iw_{m}\left(s\right)}^{\prime}\left(r\right)\right\rangle_{\kappa}$
$\displaystyle=$ $\displaystyle-\left\langle
u_{r},\tfrac{D}{dr}\left.\tfrac{d}{ds}\right|_{0}\gamma_{iw_{m}\left(s\right)}\left(r\right)\right\rangle_{\kappa}=-\left\langle
u_{r},\tfrac{D}{dr}K_{m}\left(r\right)\right\rangle_{\kappa}\text{.}$

where $K_{m}$ is the vector field along $\gamma_{v}$ defined in (15). By (16),
we have

$-\left\langle
u_{r},K_{m}^{\prime}\left(r\right)\right\rangle_{\kappa}=\begin{cases}\kappa
s_{\kappa}(r),&\text{if }m=1\\\ 0,&\text{if }m=2\\\ c_{\kappa}(r),&\text{if
}m=3.\end{cases}$

In the same way,

$\left\langle
Y_{m}\left(r\right),n_{r}\right\rangle_{\kappa}=\left\langle\left.\tfrac{D}{ds}\right|_{0}W_{m}\left(r,s\right),N_{m}\left(r,0\right)\right\rangle_{\kappa}=-\left\langle
W_{m}\left(r,0\right),\left.\tfrac{D}{ds}\right|_{0}N_{m}\left(r,s\right)\right\rangle_{\kappa}=-\zeta_{m}\left(r\right)\text{,}$

with $\zeta_{m}$ as in Lemma 12.

For $\ell\in\mathcal{M}$, unless otherwise stated, we consider the
parametrization $\gamma$ with $\gamma(0)\in S$ and $[\gamma]=\ell$. The next
lemma refers to the isomorphism defined in (7).

###### Lemma 14

The tangent space $T_{\left[\gamma\right]}\mathcal{M}$ identifies with
$\left\\{K\in\mathfrak{J}_{\gamma}\mid K\left(0\right)\in
T_{\gamma\left(0\right)}S\right\\}$ via the isomorphism $T_{\gamma}$.

Proof. Let $X\in T_{[\gamma]}\mathcal{M}$ and let $c$ be a smooth curve on
$\mathcal{M}$ (defined on an interval $I$ containing 0) such that
$c(0)=[\gamma]$ and $c^{\prime}(0)=X$. For each $t\in I$, let
$[\gamma_{t}]\in\mathcal{M}$ such that $\gamma_{t}(0)\in S$ and
$[\gamma_{t}]=c(t)$. Hence, by (8), $X=T_{\gamma}(J^{N})$, where $J$ is given
by

$J(s)=\left.\tfrac{d}{dt}\right|_{0}\gamma_{t}(s).$

Now, $t\mapsto\gamma_{t}(0)$ is a smooth curve on $S$. Then
$J(0)=\left.\frac{d}{dt}\right|_{0}\gamma_{t}(0)\in T_{\gamma(0)}S$ and so
$J^{N}(0)\in T_{\gamma(0)}S$, since $\gamma^{\prime}\left(0\right)\in
T_{\gamma(0)}S$.

Thus, $T_{[\gamma]}\mathcal{M}\subset
T_{\gamma}(\left\\{K\in\mathfrak{J}_{\gamma}\mid K\left(0\right)\in
T_{\gamma\left(0\right)}S\right\\})$. The proof concludes noting that both
spaces have dimension 3.

Proof of Theorem 3. For $\kappa=0,-1$, let $T=\infty$ and for $\kappa=1$, let
$T=\pi/2$. We consider

$F:\mathcal{M}\times(-T,T)\cong
T^{1}S\times(-T,T)\rightarrow\mathcal{G}_{\kappa}$

defined by $F(u,t)=\left[\gamma_{u_{t}}\right]$, where $u_{t}$ is the parallel
transport of $u$ between $0$ and $t$ along $\gamma_{iu}.$ Clearly, $F$ is a
smooth function. Now, we will prove that condition a) of Definition 2 holds.
Via the identification of $\mathcal{M}$ with $T^{1}S$, given $u\in T^{1}S$ we
consider the curve $t\mapsto\alpha_{u}(t)=F(u,t)$, which satisfies that
$\alpha_{u}\left(0\right)=[\gamma_{u}]$. Besides, the Jacobi field along
$\gamma_{u}$ given by

$J(s)=\left.\tfrac{d}{dt}\right|_{0}\gamma_{u_{t}}(s)$

holds $J(0)=iu\in T_{\gamma_{u}(0)}S$. Therefore,
$J\in\mathfrak{J}_{\gamma_{u}}$ and by Lemma 14, $\alpha_{u}^{\prime}(0)\in
T_{[\gamma_{u}]}\mathcal{M}$.

The fact that $F_{+}$ and $F_{-}$ are bijections onto $\mathcal{U}$ and that
the billiard map determined by $\left(F,\mathcal{U},\mathcal{M}\right)$
coincides with the outer billiard map on $\mathcal{G}_{\kappa}$ associated
with $S$ follow directly from the definitions. So, to verify condition b), it
remains to prove that $F_{+}$ and $F_{-}$ are diffeomorphisms. For this we
have to check that

$dF_{\left(u,t\right)}:T_{\left(\ell,t\right)}\left(\mathcal{M}\times(-T,T)\right)\cong
T_{u}T^{1}S\times T_{t}\mathbb{R}\rightarrow
T_{F\left(u,t\right)}\mathcal{G}_{\kappa}$

is non singular for all $u\in T^{1}S$ and $0\neq\left|t\right|<T$, where
$\ell=[\gamma_{u}]$.

We consider the basis $\mathcal{B}$ of
$T_{\left(\ell,t\right)}\left(\mathcal{M}\times(-T,T)\right)\cong
T_{u}T^{1}S\times T_{t}\mathbb{R}$ given by

$\mathcal{B}=\left\\{(w_{1}^{\prime}(0),0),(w_{2}^{\prime}(0),0),(w_{3}^{\prime}(0),0),\left(0,\left.\tfrac{d}{ds}\right|_{t}\right)\right\\},$

where $w_{m}$ are the curves in $T^{1}S$ as in Lemma 10. For $m=1,2,3$ we have

$(dF)_{(u,t)}(w_{m}^{\prime}(0),0)=\left.\tfrac{d}{ds}\right|_{0}\alpha_{w_{m}(s)}(t)$

and

$(dF)_{(u,t)}\left(0,\left.\tfrac{d}{ds}\right|_{t}\right)=\left.\tfrac{d}{ds}\right|_{0}\alpha_{u}(t+s)\text{.}$

By (8), these vectors correspond to the Jacobi fields
$J_{m}\in\mathfrak{J}_{\gamma_{u_{t}}}$ whose initial conditions are (using
the notation given immediately before Lemma 11)

$J_{4}(0)=v_{t}\hskip 14.22636pt\text{and}\hskip
14.22636ptJ_{4}^{\prime}(0)=0\text{,}$

and for $m=1,2,3$,

$J_{m}(0)=K_{m}(t)-\left\langle
K_{m}(t),u_{t}\right\rangle_{\kappa}u_{t}\hskip 14.22636pt\text{and}\hskip
14.22636ptJ_{m}^{\prime}(0)=\left.\tfrac{D}{ds}\right|_{0}W_{m}\left(t,s\right)=Y_{m}\left(t\right),$

where $K_{m}$ is as in (15) and $W_{m}$ is defined in Lemma 13.

Now, we consider the basis

$\mathcal{B}_{t}=\\{E_{1}^{t},\cdots,E_{4}^{t}\\}$ (17)

of $\mathfrak{J}_{\gamma_{u_{t}}}$, where $E_{m}^{t}$ are the Jacobi fields
along $\gamma_{u_{t}}$ whose initial conditions are

$\begin{array}[]{cccc}E_{1}^{t}(0)=0\text{,}&E_{2}^{t}(0)=n_{t}\text{,}&E_{3}^{t}(0)=0\text{,}&E_{4}^{t}(0)=v_{t}\text{,}\\\
&&&\\\
(E_{1}^{t})^{\prime}(0)=n_{t}\text{,}&(E_{2}^{t})^{\prime}(0)=0\text{,}&(E_{3}^{t})^{\prime}(0)=v_{t}\text{,}&(E_{4}^{t})^{\prime}(0)=0\text{.}\end{array}$
(18)

By Lemmas 11 and 13, we have

$J_{m}(0)=\begin{cases}b_{21}s_{\kappa}(t)n_{t},&m=1\\\
v_{t}+b_{22}s_{\kappa}(t)n_{t},&m=2\\\ 0,&m=3\\\ v_{t},&m=4\end{cases}\hskip
14.22636pt\text{and}\hskip 14.22636ptJ_{m}^{\prime}(0)=\begin{cases}\kappa
s_{\kappa}(t)v_{t}+b_{11}n_{t},&m=1\\\ b_{12}n_{t},&m=2\\\
c_{\kappa}(t)v_{t},&m=3\\\ 0,&m=4.\end{cases}$

Hence, calling $C$ the matrix of $(dF)_{(u,t)}$ with respect to the bases
$\mathcal{B}$ and $\mathcal{B}_{t}$, we obtain that

$C=\left(\begin{array}[]{cccc}b_{11}&b_{12}&0&0\\\
b_{21}s_{\kappa}(t)&b_{22}s_{\kappa}(t)&0&0\\\ \kappa
s_{\kappa}(t)&0&c_{\kappa}(t)&0\\\ 0&1&0&1\end{array}\right).$

Therefore, $\det C=c_{\kappa}(t)s_{\kappa}(t)b$, where
$b=\det[A_{p}]_{\\{u,v\\}}$. Since, by hypothesis, $0\neq|t|<T$ and
$[A_{p}]_{\\{u,v\\}}$ is definite, we have that $C$ is non singular.

Proof of Theorem 5. By the Definition 2 of the outer billiard
$B:\mathcal{U}\rightarrow\mathcal{U}$ we have that

$B=F_{+}\circ g\circ F_{-}^{-1},$

where $g:T^{1}S\times(-T,0)\rightarrow T^{1}S\times(0,T)$ is defined by
$g(u,t)=(u,-t)$.

Given $\ell\in\mathcal{U}$, suppose that $\ell=F_{-}(u,-t)$ for some $0<t<T$ .
We compute the matrix of $dB_{\ell}$ with respect to the canonical bases
$\mathcal{B}_{-t}$ and $\mathcal{B}_{t}$ of $\mathfrak{J}_{\gamma_{u_{-t}}}$
and $\mathfrak{J}_{\gamma_{u_{t}}}$ as in (18), respectively, obtaining

$\left[dB_{\ell}\right]_{\mathcal{B}_{-t},\mathcal{B}_{t}}=\left(\begin{array}[]{cc}R&0_{2}\\\
D&R\end{array}\right)\text{,}$ (19)

where $R=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$ and
$D=\frac{2}{b}\left(\begin{array}[]{cc}s_{\kappa}\left(t\right)\kappa
b_{22}&\kappa b_{12}\\\
-b_{21}&-\frac{b_{11}}{s_{\kappa}\left(t\right)}\end{array}\right)$, with
$b=\det\left[A_{p}\right]_{\left\\{u,v\right\\}}$. Thus,

$\begin{array}[]{c}dB_{\ell}\left(E_{1}^{-t}\right)=E_{1}^{t}+\frac{2}{b}s_{\kappa}\left(t\right)\kappa
b_{22}E_{3}^{t}-\frac{2}{b}b_{21}E_{4}^{t},\\\ \\\
dB_{\ell}\left(E_{2}^{-t}\right)=-E_{2}^{t}+\frac{2}{b}\kappa
b_{12}E_{3}^{t}-\frac{2}{b}\frac{b_{11}}{s_{\kappa}\left(t\right)}E_{4}^{t},\end{array}$
$dB_{\ell}\left(E_{3}^{-t}\right)=E_{3}^{t}\hskip 7.11317pt\text{and}\hskip
7.11317ptdB_{\ell}\left(E_{4}^{-t}\right)=-E_{4}^{t}.$

Recall from (13) the definition of the symplectic form $\omega_{K}$.
Straightforward computations yield that $\left\langle
E_{i}^{t}\left(0\right)\times
E_{j}^{t}\left(0\right),u_{t}\right\rangle_{\kappa}=\left\langle\left(E_{i}^{t}\right)^{\prime}\left(0\right)\times\left(E_{j}^{t}\right)^{\prime}\left(0\right),u_{t}\right\rangle_{\kappa}=0$
for all $1\leq i<j\leq 3$ except for

$\left\langle E_{2}^{t}\left(0\right)\times
E_{4}^{t}\left(0\right),u_{t}\right\rangle_{\kappa}=\left\langle\left(E_{1}^{t}\right)^{\prime}\left(0\right)\times\left(E_{3}^{t}\right)^{\prime}\left(0\right),u_{t}\right\rangle_{\kappa}=-1$

Hence,

$\left[\omega_{K}\right]_{\mathcal{B}_{t}}=\left(\begin{array}[]{cc}0_{2}&\rho\\\
-\rho&0_{2}\end{array}\right)$

with $\rho=\left(\begin{array}[]{cc}\kappa&0\\\ 0&1\end{array}\right)$. We
observe that
$\left[\omega_{K}\right]_{\mathcal{B}_{-t}}=\left[\omega_{K}\right]_{\mathcal{B}_{t}}$.
Hence, calling $H$ the matrix in (19), we have to check that

$H^{T}\left[\omega_{K}\right]_{\mathcal{B}_{t}}H=\left[\omega_{K}\right]_{\mathcal{B}_{-t}}$

The left hand side equals

$\left(\begin{array}[]{cc}-D^{T}\rho R+R\rho D&\rho\\\
-\rho&0_{2}\end{array}\right)$

and

$-D^{T}\rho R+R\rho D=\frac{2}{\vskip 12.0pt plus 4.0pt minus
4.0pta}\left(1-\kappa^{2}\right)\left(\begin{array}[]{cc}0&-b_{12}\\\
b_{21}&0\end{array}\right),$

which is the zero matrix since $\kappa=\pm 1$, as desired.

Proof of Proposition 6. Following the computations in the proof of Theorem 5,
we have

$\left[\omega_{\times}\right]_{\mathcal{B}_{t}}=\left(\begin{array}[]{cc}j&0_{2}\\\
0_{2}&j\end{array}\right),$

where $j=\frac{1}{2}\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$
and
$\left[\omega_{\times}\right]_{\mathcal{B}_{-t}}=\left[\omega_{\times}\right]_{\mathcal{B}_{t}}$.
Further computations yield

$H^{T}\left[\omega_{\times}\right]_{\mathcal{B}_{t}}H=\left(\begin{array}[]{cc}j&-(RjD)^{T}\\\
RjD&j\end{array}\right).$ (20)

Now,
$RjD=\frac{1}{b}\left(\begin{array}[]{cc}-b_{21}&-b_{11}/s_{\kappa}\left(t\right)\\\
s_{\kappa}\left(t\right)\kappa b_{22}&\kappa b_{12}\end{array}\right)$ and
$b_{11}/s_{\kappa}(t)\neq 0$ since by the hypothesis $S$ is quadratically
convex. Therefore, $RjD\neq 0_{2}$ and the expression (20) is not equal to
$\left[\omega_{\times}\right]_{\mathcal{B}_{-t}}$.

In order to prove Theorem 4 we need the presentation of $\mathcal{G}_{\kappa}$
as a symmetric homogeneous space. The details of the following description can
be found for instance in [9]. For $\kappa=\pm 1$, we consider the standard
presentation of $M_{\kappa}$ as a submanifold of $\mathbb{R}^{4}$: If
$\\{e_{0},e_{1},e_{2},e_{3}\\}$ is the canonical basis of $\mathbb{R}^{4}$,
then $M_{k}$ is the connected component of $e_{0}$ of the set

$\left\\{x\in\mathbb{R}^{4}\,|\,\kappa
x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=\kappa\right\\}.$

Let $G_{\kappa}$ be the identity component of the isometry group of
$M_{\kappa}$, that is, $G_{1}=SO_{4}$ and $G_{-1}=O_{o}\left(1,3\right)$. The
group $G_{\kappa}$ acts smoothly and transitively on $\mathcal{G}_{\kappa}$ as
follows: $g\cdot[\gamma]=[g\circ\gamma]$. Let $\gamma_{o}$ be the geodesic in
$M_{\kappa}$ with $\gamma_{o}(0)=e_{0}$ and initial velocity $e_{1}\in
T_{e_{0}}M_{\kappa}$ and let $H_{\kappa}$ be the isotropy subgroup of
$G_{\kappa}$ at $[\gamma_{o}]$. Then there exists a diffeomorphism
$\phi:G_{\kappa}/H_{\kappa}\rightarrow\mathcal{G}_{\kappa}$, given by
$\phi(gH_{\kappa})=g\cdot[\gamma_{o}]$.

The Killing form of Lie $(G_{\kappa})$ provides $G_{\kappa}$ with a bi-
invariant metric and thus there exists a unique pseudo-Riemannian metric
$\tilde{g}_{K}$ on $G_{\kappa}/H_{\kappa}$ such that the canonical projection
$\pi:G_{\kappa}\rightarrow G_{\kappa}/H_{\kappa}$ is a pseudo-Riemannian
submersion. The diffeomorphism $\phi$ turns out to be an isometry onto
$\mathcal{G}_{\kappa}$ endowed with a constant multiple of the metric $g_{K}$
defined in (11).

Besides, it is well known that $(G_{\kappa}/H_{\kappa},\tilde{g}_{K})$ is a
pseudo-Riemannian symmetric space. In particular, if
Lie$\,\left(G_{\kappa}\right)=$
Lie$\,\left(H_{\kappa}\right)\oplus\mathfrak{p}_{\kappa}$ is the Cartan
decomposition determined by $[\gamma_{o}]$, then for any
$Z\in\mathfrak{p}_{\kappa}$ the curve $t\mapsto\exp\left(tZ\right)H_{\kappa}$
is a geodesic of $G_{\kappa}/H_{\kappa}$.

Proof of Theorem 4. We fix $u\in T_{p}^{1}S$ and we want to see that the curve
$\Gamma\left(t\right)=[\gamma_{u_{t}}]$ is the geodesic in
$(\mathcal{G}_{\kappa},g_{K})$ with initial velocity
$\mathcal{J}\left(N\left(u\right)\right)$. First we verify that
$\Gamma^{\prime}\left(0\right)=\mathcal{J}\left(N\left(u\right)\right)$ and
afterwards that $\Gamma$ is a geodesic of $\mathcal{G}_{\kappa}$.

The initial velocity of $\Gamma$ corresponds, via the isomorphism
$T_{\gamma_{u}}$ of (7), with the Jacobi field along $\gamma_{u}$ given by

$J(s)=\left.\tfrac{d}{dt}\right|_{0}\gamma_{u_{t}}(s)\text{.}$

A straightforward computation shows that it is determined by the conditions
$J(0)=iu$ and $J^{\prime}(0)=0$.

On the other hand, if $N$ is the unit normal vector field on $\mathcal{M}$
pointing outwards, then, after the identification with Jacobi fields,

$N([\gamma_{u}])=I\in\mathfrak{J}_{\gamma_{u}}\hskip
14.22636pt\text{satisfying}\hskip
14.22636ptI(0)=-n(p)\,\,\text{and}\,\,I^{\prime}(0)=0\text{.}$

Indeed, by (11),

$g_{K}\left(I,I\right)=\langle I,I\rangle_{\kappa}+\kappa\langle
I^{\prime},I^{\prime}\rangle_{\kappa}=\left(-1\right)^{2}\left|n(p)\right|_{\kappa}^{2}=1$

and $g_{K}(I,K)=-\langle n(p),K\left(0\right)\rangle_{\kappa}=0$ for all $K\in
T_{\left[\gamma_{u}\right]}\mathcal{M}$, since $K\left(0\right)\in
T_{\gamma_{u}\left(0\right)}S$ by Lemma 14.

Now, by the definition of the complex structure $\mathcal{J}$ in (9), the
identity
$\mathcal{J}_{[\gamma_{u}]}\left(N_{[\gamma_{u}]}\right)=\Gamma^{\prime}\left(0\right)$
translates into $J=\gamma_{u}^{\prime}\times I$, and this is true since
$J^{\prime}\left(0\right)=0=I^{\prime}\left(0\right)$ and

$J\left(0\right)=iu=n\left(p\right)\times
u=\gamma_{u}^{\prime}\left(0\right)\times I\left(0\right)\text{.}$

Next we show that $\Gamma$ is a geodesic. By homogeneity, we may suppose that
$p=e_{0}$, the inward pointing unit normal vector of $S$ at $e_{0}$ is $e_{3}$
and $u=e_{1}$. Hence $iu=e_{2}$ and $u_{t}=e_{1}\in
T_{\gamma\left(t\right)}M_{\kappa}$ where
$\gamma\left(t\right)=c_{\kappa}\left(t\right)~{}e_{0}+s_{\kappa}\left(t\right)~{}e_{2}$.

Let $Z$ be the linear transformation of $\mathbb{R}^{4}$ defined by
$Z\left(e_{0}\right)=e_{2}$, $Z\left(e_{2}\right)=-\kappa e_{0}$ and
$Z\left(e_{1}\right)=Z\left(e_{3}\right)=0$. It is easy to verify that $Z\in$
Lie$\,\left(G_{\kappa}\right)$ and
$\exp\left(tZ\right)\left[\gamma_{u}\right]=\left[\gamma_{u_{t}}\right]$ for
all $t$. Now, one can see in the preliminaries of [9] (page 752) that
$Z\in\mathfrak{p}_{\kappa}$, and so $\Gamma$ is a geodesic by the properties
of symmetric spaces presented above.

Before proving Proposition 8, we comment on the Klein model of hyperbolic
space, that is, the open ball $\mathcal{H}$ centered at the origin with radius
$1$, where the trajectories of geodesics are the intersections of Euclidean
straight lines with the ball. The intersections of $\mathcal{H}$ with
Euclidean planes are totally geodesic hyperbolic planes.

We recall the following well-known constructions on $\mathcal{H}$ (see for
instance Chapter 6 of [3]). For an oriented line $\ell$ in $\mathcal{H}$ we
call $\ell^{+}$ and $\ell^{-}$ its forward and backward ideal end points in
the two sphere $\partial\mathcal{H}$.

Let $\ell_{1}$ and $\ell_{2}$ be two coplanar oriented lines in $\mathcal{H}$
such that the corresponding extensions to Euclidean straight lines intersect
in the complement of the closure of $\mathcal{H}$. In particular, $\ell_{1}$
and $\ell_{2}$ do not intersect and are not asymptotic and hence there exists
the shortest segment joining them with respect to the hyperbolic metric; we
call it $s\left(\ell_{1},\ell_{2}\right)$.

The hyperbolic midpoint of $s\left(\ell_{1},\ell_{2}\right)$ is the
intersection of the Euclidean segments joining $\ell_{1}^{+}$ with
$\ell_{2}^{-}$ and $\ell_{1}^{-}$ with $\ell_{2}^{+}$, or joining
$\ell_{1}^{+}$ with $\ell_{2}^{+}$ and $\ell_{1}^{-}$ with $\ell_{2}^{-}$,
depending on the orientation of the lines.

Suppose that $\ell_{1},\ell_{2}$ lie in the plane $P$ and call $D$ the disc
$P\cap\mathcal{H}$ and $C$ its boundary. Now we describe the construction of
the segment $s\left(\ell_{1},\ell_{2}\right)$ itself. We will need it only in
the case when one of the lines is a diameter in $D$, say $\ell_{1}$. Let $p$
be the intersection of the tangent lines to $C$ through the ideal end points
$\ell_{2}^{+}$ and $\ell_{2}^{-}$. Then $s\left(\ell_{1},\ell_{2}\right)$ is
the segment joining $\ell_{1}$ and $\ell_{2}$ contained in the straight line
through $p$ perpendicular to $\ell_{1}$ (the point $p$ is called the pole of
$\ell_{2}$ in the plane $P$).

We recall the formula for the hyperbolic distance between a point in
$\mathcal{H}$ and the midpoint of any chord containing it: Let $x,y$ be two
distinct points in $S^{2}=\partial\mathcal{H}$ and let $c$ be the midpoint of
the segment joining $x$ and $y$, that is, $c=\frac{1}{2}\left(x+y\right)$.
Then, for any $t\in\left(0,1\right)$,

$d\left(c,c+t\,\tfrac{y-x}{2}\right)=\text{arctanh~{}}t$ (21)

holds, where $d$ is the hyperbolic distance in $\mathcal{H}$.

Since quadratical contact is invariant by diffeomorphisms, a quadratically
convex surface of $\mathbb{R}^{3}$ contained in $\mathcal{H}$ is also
quadratically convex with the hyperbolic metric. By abuse of notation, we
describe an oriented geodesic $\ell$ in $\mathcal{H}$ by the straight
Euclidean $p+\mathbb{R}u$ line containing $\ell$, with $p\in\mathcal{H}$ and
$u$ a unit vector giving the orientation.

Proof of Proposition 8. We use the Klein model of hyperbolic space. Let
$\ell=\mathbb{R}e_{3}$ and
$\ell_{\theta}=\mathbb{R}\left(\sin\theta~{}e_{2}+\cos\theta~{}e_{3}\right)$.
We construct a surface $S$ contained in the region $x\geq 0,y\geq 0$ of
$\mathcal{H}$ whose associated outer billiard map $B$ satisfies
$B^{3}\left(\ell\right)=\ell_{\theta}$. We fix a real number $r$ in the
interval $\left(\sin\theta,1\right)$. Call $\ell_{1}=re_{1}+\mathbb{R}e_{3}$,
$\ell_{2}=re_{2}+\mathbb{R}e_{3}$ and let $p=\left(x_{o},0,0\right)$ and
$q=\left(0,y_{o},z_{o}\right)$ be the hyperbolic midpoints between $\ell$ and
$\ell_{1}$ and between $\ell_{\theta}$ and $\ell_{2}$, respectively.

Figure 4: Elements for the construction of $S$

There exists a quadratically convex compact surface $S$ contained in
$\mathcal{H}$ tangent to the vertical planes $y=0$, $y+x=r$ and $x=0$, at the
points $p,$ $\left(r/2,r/2,0\right)$ and $q$, respectively. In fact, consider
a quadratically convex compact surface $S^{\prime}$ tangent to those planes at
the points $p,$ $\left(r/2,r/2,0\right)$ and $\left(0,y_{o},0\right)$,
respectively, such that the absolute value of the height function
$\left.z\right|_{S^{\prime}}$ is bounded by $\varepsilon$ for some
$\varepsilon>0$. Let $T$ be the unique affine transformation of
$\mathbb{R}^{3}$ fixing the vertical planes through $p$ and
$\left(r/2,r/2,0\right)$ and sending $\left(0,y_{o},0\right)$ to $q$. Then
$S=T\left(S^{\prime}\right)$ satisfies the desired conditions, provided that
$\varepsilon$ is small enough (notice that affine transformations preserve
quadratical contact). By the properties of $S$ we have that
$B\left(\ell\right)=\ell_{1}$, $B^{2}\left(\ell\right)=\ell_{2}$ and
$B^{3}\left(\ell\right)=\ell_{\theta}$.

Proof of Proposition 9. As in the proof of Proposition 8, we use the Klein
model $\mathcal{H}$ for hyperbolic space. We write
$\mathbb{R}^{3}=\mathbb{C}\times\mathbb{R}$. Given $0<a<\frac{1}{2}<$
$r_{o}<1$ and $h_{o}=\sqrt{1-r_{o}^{2}}$, we consider the straight lines

$\begin{array}[]{ll}\gamma_{0}\left(t\right)=\left(t,0\right)\text{,}&\gamma_{1}\left(t\right)=\left(\tfrac{i}{2}+t\left(1-ai\right),0\right)\text{,}\\\
\gamma_{2}\left(t\right)=\left(\tfrac{i}{2}+t\left(1-ai\right),h_{o}\right)\text{,}&\gamma_{3}\left(t\right)=\left(t,h_{o}\right)\text{.}\end{array}$

For $k=0,1,2,3$, let $\ell_{k}$ be the corresponding oriented geodesic in
$\mathcal{H}$ and set $\ell_{4}=\ell_{0}$. They are pairwise coplanar. We call
$P_{k}$ the hyperbolic plane containing $\ell_{k}$ and $\ell_{k+1}$ and
$\sigma_{k}$ the shortest segment joining them (they are not asymptotic since
$0<2a<1$).

We will show the existence of a quadratically convex compact surface $S$ in
$\mathcal{H}$ not intersecting $\ell_{0}$ such that the associated billiard
map $B$ satisfies $B^{k}\left(\ell_{0}\right)=\ell_{k}$ for $k=0,\dots,4$ and
its holonomy at $\ell_{0}$ is not trivial.

We make computations for general values of $r$ and $h=\sqrt{1-r^{2}}$, with
$0<a<\frac{1}{2}<r\leq 1$, in order to deal simultaneously with $\ell_{0}$ and
$\ell_{1}$ on the one hand (case $r=1$) and $\ell_{2}$ and $\ell_{3}$ on the
other (case $r=r_{o}$), since the former lie in a disc of radius $1$ at height
$0$ and the latter in a disc of radius $r_{o}$ at height $h_{o}$.

The end points of $\ell_{1}$ and $\ell_{2}$ are given by

$\ell_{1}^{\varepsilon}=\left(z_{\varepsilon}\left(1\right),0\right)\text{\ \
\ \ \ \ and\ \ \ \ \ \
}\ell_{2}^{\varepsilon}=\left(z_{\varepsilon}\left(r_{o}\right),h_{o}\right)\text{,}$

for $\varepsilon=\pm 1$, where
$z_{\varepsilon}\left(r\right)=\frac{i}{2}+t_{\varepsilon}\left(r\right)\left(1-ai\right)$,
with $t_{-}\left(r\right)<t_{+}\left(r\right)$ being the solutions of the
equation $t^{2}+\left(\frac{1}{2}-at\right)^{2}=r^{2}$. Since $\ell_{1}$ and
$\ell_{2}$ are parallel, the end points of $\sigma_{1}$ are the respective
midpoints, whose (common) component in $\mathbb{C}$ is

$z_{o}=\tfrac{1}{2}\left(z_{+}\left(1\right)+z_{-}\left(1\right)\right)=\tfrac{1}{2}\left(z_{+}\left(r_{o}\right)+z_{-}\left(r_{o}\right)\right)=\tfrac{1}{2\left(a^{2}+1\right)}\left(a+i\right)\text{.}$

Hence, $q_{+}\left(\ell_{1}\right)=\left(z_{o},0\right)$ and
$q_{-}\left(\ell_{2}\right)=\left(z_{o},h_{o}\right)$. Similarly,
$q_{-}\left(\ell_{0}\right)=\left(0,0\right)$ and
$q_{+}\left(\ell_{3}\right)=\left(0,h_{o}\right)$.

Figure 5: Elements for the construction of $S$

Let $p_{1}=\left(w\left(1\right),0\right)$ and
$p_{2}=\left(w\left(r_{o}\right),h_{o}\right)$ be the poles of the line
$\ell_{1}$ in the plane $\mathbb{C}\times\left\\{0\right\\}$ and of the line
$\ell_{2}$ in the plane $\mathbb{C}\times\left\\{h_{o}\right\\}$,
respectively. That is, $w\left(r\right)$ is the intersection of the lines
tangent to the circle of radius $r$ in $\mathbb{C}$ at the points
$z_{-}\left(r\right)$ and $z_{+}\left(r\right)$. Using the construction of the
shortest segment joining two oriented lines, the segment $\sigma_{2}$ is
contained in the line perpendicular to $\ell_{3}$ passing through
$\left(w\left(r_{o}\right),h_{o}\right)$, that is, the line
$\left(\operatorname{Re}w\left(r_{o}\right)+\mathbb{R}i,h_{o}\right)$. Putting
$r=1$, we get that $\sigma_{0}$ is contained in the line
$\left(\operatorname{Re}w\left(1\right)+\mathbb{R}i,0\right)$.

We have that
$w\left(r\right)=z_{+}\left(r\right)+s_{o}\left(r\right)iz_{+}\left(r\right)$,
where $s_{o}$ is the solution of the equation

$z_{+}\left(r\right)+siz_{+}\left(r\right)=z_{-}\left(r\right)-siz_{-}\left(r\right)\text{.}$

A straightforward computation yields
$\operatorname{Re}w\left(r\right)=2ar^{2}$.

Now, computing the intersections of the remaining $\sigma_{k}$ with the lines
$\ell_{j}$, we obtain the rest of the $q_{\delta}\left(\ell_{j}\right)$,
$\delta=\pm 1$:

$\begin{array}[]{ll}q_{+}\left(\ell_{0}\right)=\left(\operatorname{Re}w\left(1\right),0\right)=2\left(a,0\right)\text{,}&q_{-}\left(\ell_{1}\right)=2\left(a+i\left(\frac{1}{4}-a^{2}\right),0\right)\\\
q_{+}\left(\ell_{2}\right)=\left(2\left(ar_{o}+i\left(\frac{1}{4}-a^{2}r_{o}^{2}\right)\right),h_{o}\right)\text{,}&q_{-}\left(\ell_{3}\right)=\left(\operatorname{Re}w\left(r_{o}\right),h_{o}\right)=\left(2ar_{o}^{2},h_{o}\right)\text{.}\end{array}$

As in Proposition 8, for each $a>0$ there exists a quadratically convex
surface $S_{a}$ in $\mathcal{H}$ tangent to the plane $P_{k}$ at the midpoint
of $\sigma_{k}\,$for any $k=0,\dots,3$. The associated billiard map $B_{a}$
satisfies $\left(B_{a}\right)^{4}\left(\ell_{0}\right)=\ell_{0}$ and its
holonomy at $\ell_{0}$ turns out to be

$H\left(a\right)=\sum\nolimits_{k=0}^{3}\left(-1\right)^{k}d\left(q_{+}\left(\ell_{k}\right),q_{-}\left(\ell_{k}\right)\right).$

Particularizing $r_{o}=h_{o}=1/\sqrt{2}$, using (21) we obtain that

$H\left(a\right)=\text{arctanh}\left(2a\right)-\text{arctanh}\left(a\sqrt{4a^{2}+3}\right)+\text{arctanh}\left(a\sqrt{2a^{2}+1}\right)-\text{arctanh}\left(a\right)\text{.}$

We compute $H\left(0\right)=0$ and $H^{\prime}\left(0\right)\neq 0$. Therefore
for a small enough positive $a$, the holonomy of $B_{a}$ at $\ell_{0}$ does
not vanish.

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Yamile Godoy
Conicet - Universidad Nacional de Córdoba
CIEM - FaMAF
Ciudad Universitaria, 5000 Córdoba, Argentina
<EMAIL_ADDRESS>
Michael Harrison
Institute for Advanced Study
1 Einstein Drive
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Marcos Salvai
Conicet - Universidad Nacional de Córdoba
CIEM - FaMAF
Ciudad Universitaria, 5000 Córdoba, Argentina
<EMAIL_ADDRESS>