Lecture Notes for 120 - UCLA Department of Mathematics

7.6. THE UPPER HALF PLANE 164
Proof. Since F is an isometry and it preserves q we must also have that it
preserves its velocity and tangential acceleration
DF ( ˙q (t)) = ˙q (t) ,
DF ¨q I (t) = ¨q I (t) .
As q is unit speed we have ˙q · ¨q I =0. If ¨q I (t) 6= 0, then DF preserves q (t) as well
as the basis ˙q (t) , ¨q I (t) for the tangent space at q (t) . By the uniqueness result
above this shows that F is the identity map as that map is always a isometry that
fixes any point and basis. But this contradicts that F is nontrivial.
⇤
Note that circles in the plane are preserved by rotations, but they are not fixed,
nor are they geodesics. The picture we should have in mind for such isometries and
geodesics is a mirror symmetry in a line, or a mirror symmetry in a great circle on
the sphere.
There are some further surprises along these lines.
Theorem 7.5.4. If all geodesics are preserved by at least one non-trivial isometry,
then the space has constant Gauss curvature. Conversely, if the space has
constant Gauss curvature, then all geodesic will be fixed by some isometry.
Below we shall construct constant Gauss curvature spaces, and show that the
isometries and geodesics have these properties. More generally one will have to show
that locally all constant Gauss curvature spaces can be accounted for by knowing
only these examples.
For now lets us discuss the isometries of the plane and sphere.....
7.6. The Upper Half Plane
A particularly interesting case to study is the upper half plane where we don’t
have much intuition about what might happen. This section is devoted to calculating
the symmetries, geodesics, and curvature of this space. Recall that this is an
assignment of a first fundamental form
apple 1
I= v
0 2 1
0
v 2
to the tangent space at each point p =(u, v) 2 H = {(u, v) | v>0} . We saw that
it was possible to construct a surface of revolution
✓ 1
q (t, ✓) =
t cos (✓) , 1 ◆
t sin (✓) ,h(t) ,
r
1 1
ḣ = 1
t 2 t
whose first fundamental form is
apple 1
I= t
0 2 1 .
0
t 2
This might give us a local picture of the upper half plane but it doesn’t really help
that much.
Below we shall find the isometries and geodesics by solving the equations we
have for these objects. As we shall see, even in a case where the metric is relatively
simple, this is a very difficult task.