1.7. Cone Surfaces and Translation Surfaces 13

The complex analysis point of view on a surface at first seems

rather remote from the geometric point of view discussed above, but in

fact they are quite similar. The close connection comes from the fact

that the M¨ obius transformations play a distinguished role amongst

the complex analytic functions. One example of this is the following

result, known as the Schwarz–Pick Theorem:

Theorem 1.1. Let f be a complex analytic function from the unit

disk to itself. If f is one-to-one and onto, then f is a M¨ obius trans-

formation (and hence a hyperbolic isometry).

Theorem 1.1 is part of a larger theorem, called the Poincar´e

Uniformization Theorem. The Uniformization Theorem gives a com-

plete equivalence between the Euclidean/spherical/hyperbolic geom-

etry points of view of surfaces and the Riemann surface point of view.

The proof of this result is beyond the scope of our book, but in Chap-

ter 16 we will at least explain the result and its ramifications.

1.7. Cone Surfaces and Translation Surfaces

We have mentioned several times that the octagon surface does not

“wear” Euclidean geometry as well as the square torus does, and we

have taken some pains to explain how one can profitably view the

octagon surface with hyperbolic geometry eyes. However, in Chapter

17 we come full circle and consider the octagon surface and related

surfaces from the Euclidean geometry point of view.

Suppose, as in Figure 1.2 above, we glue together the sides of

a polygon in such a way that the sides in each pair of glued sides

have the same length. The resulting surface has the property that

it is locally indistinguishable from the Euclidean plane, except at

finitely many points. At these finitely many points, a bug living in the

surface would notice some problem related to spinning around, as we

discussed above. These special points are cone points. A Euclidean

cone surface is a surface that is flat except at finitely many cone

points.

When we discussed the “torus-like” surface defined in connection

with Figure 1.2, we mentioned the spinning-around problem a bug

would face when sitting at the 2 special points. At one of the special