- INTRODUCTION TO HYPERBOLIC SPACE 173
We leave it as an exercise for the reader to verify the lemma. HINT: Recall that
an isometry is determined by its differential at a single point. Show that SL(2, JR)
acts transitively on the space of all unit tangent vectors.
We can also find all geodesics in the hyperbolic plane as follows. Since we know
a lot of isometries, it suffices to find one geodesic and then use isometries to find
others.EXAMPLE 31.3. The positive y-axis is a geodesic in the upper half-plane U^2.
Let "( : [O, 1 J ---+ U^2 be a smooth path from ia to ib, where 0 < a < b. The
length in the hyperbolic metric of "f is
{
1
b'(t)I
L("f) =Jo Im("f(t)) dt.
Write "((t) = x(t) + iy(t), where y(t) > 0. Then-1 1 vx' (t)2 + y' (t)2 111 y'(t) I- ~
L("f) - ( ) dt ?. ( ) dt - ln.0 yt 0 yt a
Note that L("f) = ln ~ if and only if x(t) = 0 and y'(t) ?. 0. This shows that
the positive y-axis is a minimal geodesic and that the distance between ia and ib
is dlU(ia, ib) = ln ~-
By using the isometrics z H ~;:~ and the fact that Mobius transformations
preserve angles and the set of all circles and lines, we obtain that all geodesics in
U^2 are (portions of) vertical lines or circles perpendicular to the x-axis.
All of the above computations can be carried out, without too much change,for the upper half-space model un for n ?. 3. The counterpart of the map z H -~
for U^2 is the inversion. Recall t hat the inversion ly,r about the sphere centered
at y E !Rn of radius r is the bijection of !Rn U { oo} defined by
2 x -y
ly,r(x ) = r Ix - Yl2 + y.Inversions of !Rn U { oo} preserve angles, i.e., are conformal. The inversion about the
unit sphere centered at the origin is lo, 1 (x ) = ~,which is the orientation-reversing
isometry z H i in dimension 2. Obviously, lQ,l preserves un.
A Mobius transformation of !Rn is defined to be a composition of the inver-
sion lo, 1 with x H >.Bx+ b, BE O(n), b E !Rn, >. > 0. We denote the group of all
Mobius transformations of !Rn by Mob(n). The subgroup of orientation-preserving
Mobius transformations is denoted by Mob+(n).
LEMMA 31.4. The inversion lo,1 preserves the hyperbolic metric 91U on un.
We leave the verification of the lemma as an exercise; this is a straightforward
generalization of the calculation that we performed for the upper half-plane above.
Consider JRn-l as the subspace JRn-l x {O} C !Rn. Then each Mobius trans-
formation T of JRn-l extends naturally to a Mobius transformation T* of !Rn so
that T* (Un) = un. Here is how it works. The extension of the inversion lo, 1 is the
inversion lo, 1 about the unit sphere in !Rn; the extension of x H >.x is given by the
same formula; and the extension of x H Ax+ b, A E O(n - 1), is given by sending
(x, t) to (Ax+ b, t) for (x, t) E JRn-l x JR. In short, the extension map produces an
injective group homomorphism ¢ from Mob( n - 1) to Mob( n) so that the image