1550251515-Classical_Complex_Analysis__Gonzalez_

242 Chapter^5

the heading of projective geometry. This made Cayley claim, with some
exaggeration, that "projective geometry is all geometry."t
A particularly simple interpretation of the Lobachevsky geometry in
the plane was made by H. Poincare in connection with his theory of
automorphic functions. In this interpretation the upper complex half-
plane H is considered to be an image of the Lobachevsky plane, the

points z, with Imz > 0, corresponding to the points of the hyper-

bolic geometry. However, the straight lines of the hyperbolic geometry

are made to correspond to the semicircles and half-lines orthogonal to

the real axis. That is, the lines in H are of two types: the semicir-

cles C = {z: lz -al= r,a E IR,r > O,Imz > O} and the vertical half-lines

L = {z: z = x 0 + iy, y > O}. The real axis is called the horizon, or the line

at infini·ty, of the hyperbolic plane (Fig. 5.15).

It is an easy matter to verify that the axioms of incidence, order, and

continuity of the hyperbolic geometry (which are the same as those of

Euclidean geometry) are satisfied in Poincare's model. For instance, it is

clear that if two semicircles do intersect, they intersect at just one point.

Also, that two points P and Q determine just one semicircle with center
on the x-axis (i.e., orthogonal to that axis), and so on.
As to the axiom of parallelism, we recall that in Euclidean geometry
given a line m and a point P without the line, there exists just one line
through P parallel to m. In hyperbolic geometry given a "line" C and
a point P, not on C, the lines through P fall into three different types:

(1) those that intersect C (as C 3 in Fig. 5.15); (2) those that meet Cat

infinity (in Poincare's model, those that are tangent to C on the x-axis)-

L

H

a

Fig. 5.15

tThe systematization of the geometries attained in the Erlangen Program does
not include the so-called Riemannian geometries. A more general approach based
on the concepts of geometric object and pseudogroup was proposed by Veblen
and Whitehead in The Foundations of Differential Geometry, Cambridge Tract
No. 29 (1932).