Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

15 Hyperbolic Geometry
S(0, 1) followed by the reflection in the sphere S(0, √ k). The transformations of the form
φ(x) =
ax + b
cx + d
where a, b, c, d are real and such that ad − bc = 0 are Möbius transformations. In fact φ(x) =
φ4 ◦ φ3 ◦ φ2 ◦ φ1(x) where φ1(x) = x + d
c is a translation, φ2(x) = 1
x is an inversion and a reflection
with respect to the real axis, φ3(x) = − ad−bc
c 2
is a translation.
(2.6)
x is a magnification and a rotation and φ4(x) = x+ a
c
In what follows we will denote with Σ either a sphere S(a, r) or a plane P (a, t) in R n .
Theorem 2.1.1. For every Möbius transformation φ(·) and every sphere or plane Σ we have that
φ(Σ) is also a sphere or a plane.
The inversive product (Σ, Σ ′ ), given in Definition 2.1.3, is a real expression which depends only
on Σ and Σ ′ and which is invariant under all Möbius transformations. When Σ and Σ ′ intersect
it is a function of their angle of intersection and when Σ and Σ ′ are disjoint it is a function of the
hyperbolic distance between them.
Since the equation defining a sphere S(a, r) is
and the equation defining a plane P (a, t) is
|x| 2 − 2〈x, a〉 + |a| 2 − r 2 = 0
−2〈x, a〉 + 2t = 0,
we have that every sphere or plane Σ can be written in a common form as
a0|x| 2 − 2〈x, a〉 + an+1 = 0,
where a = (a1, . . . , an) and the coefficient vector (a0, . . . , an+1) is determined within a real nonzero
multiple. Since for Σ = S(a, r) and Σ = P (a, t) we have that |a| 2 > a0an+1, we can give the
following definition.
Definition 2.1.3. Let Σ and Σ ′ have coefficient vectors (a0, . . . , an+1) and (b0, . . . , bn+1). The
inversive product (Σ, Σ ′ ) of Σ and Σ ′ is defined by
(Σ, Σ ′ ) =
|2〈a, b〉 − a0bn+1 − an+1b0|
2(|a| 2 − a0an+1) 1/2 (|b| 2 .
− b0bn+1)
1/2
Remark 2.1.3. It is helpful to obtain the explicit expression of (Σ, Σ ′ ) in the following three
cases.
• If Σ = S(a, r) and Σ ′ = S(b, t) then (Σ, Σ ′ ) = r2 +t 2 −|a−b| 2
2rt
• If Σ = S(a, r) and Σ ′ = P (b, t) then (Σ, Σ ′ ) = |〈a,b〉−t|
r|b| .
• If Σ = P (a, r) and Σ ′ = P (b, t) then (Σ, Σ ′ ) = |〈a,b〉|
|a||b| .
.