Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

21 Hyperbolic Geometry
since
1 +
|ip − iq|2 (p − q)2
= 1 +
2Im[ip]Im[iq] 2pq = p2 + q2 = cosh log(q/p).
2pq
But, if γ(t) = x(t) + iy(t) with t ∈ [0, 1] is any smooth curve joining z to w such that γ(1) = q and
γ(0) = p, we have that
||γ|| =
1
0
|x ′ (t) + iy ′ 1
(t)|
dt ≥
y(t)
0
In general, if we do not assume that p < q, we have that
y ′ (t)
dt = log(q/p). (2.23)
y(t)
η(ip, iq) = | log(p/q)|. (2.24)
Remark 2.3.1. Formula (2.22) can be easily generalized to the (n + 1)-dimensional case. For
z, w ∈ Hn+1 , we have
|z − w|2
cosh η(z, w) = 1 + . (2.25)
2zn+1wn+1
Remark 2.3.2. From Theorem 2.3.2 we obtain that the hyperbolic sphere with center y ∈ H n+1
and hyperbolic radius r is the Euclidean sphere with center (y1, . . . , yn, yn+1 cosh r) and radius
yn+1 sinh r. In fact {x ∈ Hn+1 : η(x, y) = r} = {x ∈ Hn+1 : cosh r = 1 + |x−y|2
} this implies
2xn+1yn+1
that a point x on the hyperbolic sphere satisfies the equations
2xn+1yn+1 cosh r = 2xn+1yn+1 + (x1 − y1) 2 + · · · + (xn+1 − yn+1) 2 ,
2xn+1yn+1 cosh r = (x1 − y1) 2 + · · · + (xn − yn) 2 + x 2 n+1 + y 2 n+1,
(yn+1 sinh r) 2 = (x1 − y1) 2 + · · · + (xn − yn) 2 + (xn+1 − yn+1 cosh r) 2 .
Remark 2.3.3. It is possible to prove that the following equations are equivalent to formula
(2.22).
sinh
cosh
tanh
η(z, w)
2
η(z, w)
2
η(z, w)
2
2 x
Starting from (2.22) and applying sinh 2
=
=
=
|z − w|
,
2(Im[z]Im[w]) 1/2
(2.26)
|z − ¯w|
,
2(Im[z]Im[w]) 1/2
z − w
z
− ¯w .
(2.27)
(2.28)
1
x
= 2 (cosh x − 1) and cosh2 2
1 = 2 (cosh x + 1) we obtain
easily (2.26) and (2.27). Formula (2.28) follows from (2.26) and (2.27). It is possible to obtain the
following explicit expression for the hyperbolic metric η
observing that
cosh log
η(z, w) = log
|z − ¯w| + |z − w|
= 1 +
|z − ¯w| − |z − w|
|z − ¯w| + |z − w|
, (2.29)
|z − ¯w| − |z − w|
2|z − w| 2
|z − ¯w| 2 |z − w|2
= 1 +
− |z − w| 2 2Im[z]Im[w] .