1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

l. INTRODUCTION TO HYPERBOLIC SPACE 175

An interesting property of inversions of JR.n is that they preserve the set of

all ( codimension 1) hyp erspheres and hyperplanes in ]Rn. Since any inversion is a

composition of Lo, 1 , a scaling kx, and conjugation by a translation of ]Rn, it suffices

to establish this property for i 0 , 1. To see this, note t hat the equation of a hyperplane

or a hypersphere in ]Rn is of the form

Ax·x+Ba·x+C=O

where A , B , C E JR, a E JRn, and a· x is the standard Euclidean inner product.

Replacing x by io ,1(x) = x~x in this equation, we obtain,

A+ Ba · x + C x · x = 0.

This is again an equation of a hypersphere or hyperplane. By taking intersections

of hyperplanes and hyperspheres , we conclude that inversion preserves the set of

all circles and lines in ]Rn.

Finally, we discuss the geodesics in un. Using the same calculation as in

Example 3 1.3, which works in a ny dimension, we see that the positive Xn-axis

is a geodesic in un. Applying the isometries of un just obtained and the basic

properties of inversion, we obtain

LEMMA 3 1.6. All geodesics in the upper half-space model un of the hyperbolic

geometry are (portions of) vertical lines or circles perpendicular to the hyperplane

aun at infinity.

PROOF. Using translations g(x) = x +a, we conclude that all vertical lines

(i.e., lines perpendicular to aun) are geodesics. Using the inversion Lo,1 and scaling

f(x) = kx fork > 0, we see that all semicircles perpendicular to aun are geodesics.

On the other hand, given any nonzero tangent vector v in un, there exists a unique
semicircle or vertical line t a ngent to v. The lemma follows. D

1.3. Types of isometries of hyperbolic space.
Isometries of lHin are classified according to the locations of t heir fixed points.
Consider the disk model of hyperbolic space, so that the ideal boundary 8lHin ~

s n- l. If r.p E Isom (lHin), then r.p extends to a homeomorphism of llr ~ lHin U 8lH!n

which is topologically a closed ball. Hence by the Brouwer fixed point theorem,

r.p has a fixed point in llr.

DEFINITION 31.7. An orientation-preserving isometry r.p of lHin is

(1) elliptic if there exists a fixed point of r.p in lH!n,

(2) parabolic if there are no fixed points of r.p in lH!n and exactly one fixed

p oint of r.p on 8lHin,

(3) hyperbolic if there are no fixed points of r.p in lHin and exactly two fixed

p oints of r.p on 8lHin.

Clearly these conditions are mutually exclusive. What is nontrivial is to show
that if r.p E Isom+ (lHin) h as at least three fixed points on 8lHin, then r.p is elliptic.
See Ratcliffe's textbook [334] for a proof.

EXAMPLE 31.8. (1) In the disk model, if r.p E SO (n , JR), then r.p is an elliptic

isometry.

(2) In the upper half-sp ace model un, given b E lRn - {O} with bn = 0, the

translation

x i-----+ x + b