Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

25 Hyperbolic Geometry
Theorem 2.4.1. Let z and w be two points in H 2 . A curve γ joining z to w satisfies
||γ|| = η(z, w)
if and only if γ is a parametrization of [z, w] as a simple curve.
Proof
Let g(·) be a η-isometry such that g(ip) = z and g(iq) = w. Let γ1(t) = x1(t) + x1(t), t ∈ [0, 1],
be any curve joining ip to iq. In view of formula (2.23) we have that ||γ1|| = η(ip, iq) if and only
if x1(t) = 0 and y ′ 1(t) > 0, that is if and only if γ1 is a parametrization of [ip, iq] as a simple
curve. In fact, if ||γ1|| = η(ip, iq) this implies that x ′ 1(t) = 0 and y ′ 1(t) > 0 that is x1(t) = 0 and
y ′ 1(t) > 0 since x1(0) = 0. On the other side, if x1(t) = 0 and y ′ 1(t) > 0, than in view of (2.23)
we immediately have ||γ1|| = η(ip, iq). The theorem follows by taking into account that g(·) is a
η-isometry and then maps hyperbolic lines in hyperbolic lines.
Theorem 2.4.2. Let z and w be two distinct points in the hyperbolic plane. Then
if and only if ζ ∈ [z, w].
η(z, w) = η(z, ζ) + η(ζ, w)
Proof
If ζ ∈ [z, w] and g(·) is such that g(z) = ip, g(ζ) = iq and g(w) = iv than in view of (2.24) we
have that η(z, w) = η(z, ζ) + η(ζ, w). On the other side if ζ /∈ [z, w] and γ is a curve made of the
segments [z, ζ] and [ζ, w], then in view of Theorem 2.4.1 we have that ||γ|| > η(z, w).
Definition 2.4.2. Let l1 and l2 be two distinct geodesics. If l1 ∩ l1 = ∅, then l1 and l2 are intersecting.
If l1 ∩ l1 = ∅, then l1 and l2 are parallel.
In the hyperbolic geometry for any given hyperbolic line l and a point w not on l, there are infinitely
many hyperbolic lines through w which are parallel to l.
For each point ζ and any hyperbolic line l the distance of ζ from l is defined by
η(ζ, l) = inf η(ζ, w).
w∈l
Theorem 2.4.3. Let l1be the unique geodesic through ζ orthogonal to l and let ζ1 be the point of
intersection between l1 and l. We have that η(ζ, l) = η(ζ, ζ1).
Proof
We may assume that l is the positive imaginary axis, then l1 = {z ∈ H2 : |ζ| = |z|}. We need to
show that η(ζ, l) = η(ζ, i|ζ|). Since each point on l is of the form z = ip, p > 0, from Theorem
2.3.2 we have that
cosh η(ζ, ip) = x2 + y2 + p2 =
2yp
|ζ|
|ζ| p
+ ≥
2y p |ζ|
|ζ|
y ,