12 1. Book Overview
• The sphere is identified with all of C ∪ ∞, via stereographic
1 − x3
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 ∪ ∞.