1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

172 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

A beautiful model of hyperbolic space is the hyperboloid model. Let !En,l
denote Minkowski space, i.e., JRn+l with the (indefinite) Lorentzian metric

dxi + · · · + dx~ - dx~+i·

The hyperboloid model is the submanifold

Hn ~ {XE JRn+l : (x, x)n,l ~xi+···+ X~ - X~+l = -1 and Xn+l > 0}

with the induced metric gJHI. Note that although the ambient metric is indefinite,
the induced metric is positive definite.
The equivalence of the above models of hyperbolic space may be seen by the
following facts.

( 1) The diffeomorphism

defined by

()

. (x1, ... ,xn)
ax=-----
. 1 + Xn+l

X - Xn+len+l

1 + Xn+l

where en+l ~ (0, ... , 0, 1), is an isometry.

(2) The diffeomorphism

defined by

is an isometry.

EXERCISE 31.1. Show that a and (3 are isometries between the models.

1.2. Geodesics and the group of isometries of hyperbolic space.

Let us begin with the 2-dimensional hyperbolic plane. Recall that the upper

half-plane U^2 = {z = x + iy E c IY = Im(z) > O} has the hyperbolic metric

dx^2 + dy^2

gv = y2

-4dz dz

(z - z)^2 ·

We can list all the orientation-preserving isometries of the metric as follows. First,

from the definition of the metric, we see easily that f (z) ~ -\z, where,\ E JR>o, and

g(z) ~ z+a, where a E JR, are isometries. Next, we claim that h(z ) ~-~preserves

the hyperbolic metric. Indeed, let w = h(z). Then,

h*( ) = 4dwdw = 4d(~)d(t) = _ 4 dzdz

gv ( w-w -) 2 ( (^1) ---= 1 ) 2 ( z-z -) 2.
z z
It is well known that each Mobius transformation
(31.1) F(z)=cz+daz + b ' where ( a c b d ) ESL(2,IR),
of the upper half-plane U^2 is a composition of f's, g 's, and h above. Therefore,
Mobius transformations z H ~::~, where a, b, c, d E JR with ad - be = 1, are
orientation-preserving isometries of the hyperbolic upper half-plane U^2.
LEMMA 31.2. All orientation-preserving isometries of U^2 are Mobius transfor-
mations of the form in (31.1).