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
obius, transformations. The 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 ∞.
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