Lecture Notes for 120 - UCLA Department of Mathematics

7.6. THE UPPER HALF PLANE 166
and the equations become
(2uvh 0 ) 2 + h +2u 2 h 0 2
h +2u 2 h 0 2uvh 0 + h +2v 2 h 0 2uvh 0 = 0.
= h 2 =(2uvh 0 ) 2 + h +2v 2 h 0 2
,
Since we just studied the case where h 0 =0, we can assume that h 0 6=0, the last
equation then reduces to
h = u 2 + v 2 h 0
showing that
r
h =
u 2 + v 2
for some constant r>0. It is then an easy matter to check that the equations in
the first line also hold. Note that this map preserves the circle of radius r centered
at (0, 0) and switches points inside the circle with points outside the circle. It is
called an inversion and is a type of mirror symmetry on the upper half plane. Note
that the regular mirror symmetries in vertical lines are also isometries of H. Note
that both mirror symmetries and inversions are their own inverses:
✓
◆
ru
F F (u, v) = F
u 2 + v 2 , rv
u 2 + v 2 ✓
r
ru
= ⇣ ⌘ 2 ⇣ ⌘ 2
ru
u 2 +v +
rv u 2 + v 2 ,
2 u 2 +v 2
1
=
(u, v)
u 2
u 2 +v
+ v2
2 u 2 +v 2
= (u, v) .
◆
rv
u 2 + v 2
Between these three types of isometries we can find all of the isometries of the
half plane. There are two key observations to be made. First, for any pair p, q 2 H
we have to find a isometry that takes p to q. This can be done using translations
and scalings. Second, for any p 2 H and direction v 2 T p H we have to find a
isometry that fixes p and whose differential is a reflections in v. This can be done
with inversions or mirror symmetries in vertical lines should v be vertical.
7.6.2. The Geodesics of H. The fact that the metric is relatively simple
allows us to compute the Christoffel symbols without much trouble
u
uu = 1 @g 2 guu uu
@u =0
v
uu =
1 @g 2 gvv uu
@v
= 1 v
v
vv = 1 @g 2 gvv vv
u
vv =
@v = 1 v
1 @g 2 guu vv
@u =0
u
uv = 1 @g 2 guu uu
@v = 1 v
v
uv = 1 2 gvv @g vv
@u =0