12 1. Book Overview

• The sphere is identified with all of C ∪ ∞, via stereographic

projection

(x1,x2,x3) →

x1

1 − x3

+

x2

1 − x3

i, (0, 0, 1) → ∞.

See §9.5 for details on stereograpic projection.

Once these identifications are made, the symmetries of the rele-

vant objects are all given by maps of the form

(1.1) z →

az + b

cz + d

, a, b, c, d ∈ C, ad − bc = 1.

We will discuss these maps in more detail in §10.1. The point ∞

is added so that when the expression in equation (1.1) looks like

“something over 0”, we define it to be ∞. Various conditions are

placed on the coeﬃcients a, b, c, d to guarantee that the relevant set—

e.g., the unit disk—is preserved by the map.

These kinds of transformations are called linear fractional, or

M¨ obius, transformations. The M¨ obius transformations are prototyp-

ical examples of complex analytic functions. These are continuous

maps from C to C which have the additional property that their

matrix of partial derivatives, at each point, is a similarity—i.e., a

rotation followed by a dilation. This constraint on the partial deriva-

tives leads to a surprisingly rich family of functions and this is the

subject of complex analysis. In Chapter 13, we will give a rapid

overview of basic complex analysis, with a view towards its applica-

tion to surfaces. In Chapter 14 and Chapter 15 we will discuss some

special complex analytic functions in detail.

Going back to our polygon gluing construction, we can view sur-

faces as being made out of pieces of C that have been glued together.

This point of view leads to the notion of a Riemann surface, as we

explain in Chapter 16. One can think of a Riemann surface as a sur-

face that “wears” C in the same seamless way that the square torus

“wears” Euclidean geometry or the octagon surface “wears” hyper-

bolic geometry. Once we have the notion of a Riemann surface, we

can “do complex analysis on it” in much the same way that one can

do complex analysis in C or in C ∪ ∞.