Differential Structure of Two Dimensional Manifold

Question

I am a beginner in Differential Topology, and have learnt that

Theorem One dimensional differential manifold without boundary is diffeomorphic to $\mathbb R$ or $\mathbb S$.

• So I am curious about the classification of two dimensional differential manifold. Is it same as the topological situation?

• In even dimension, we can consider the complex situation. That is to say, how do we classify the compact Riemann surface?

Any help in thanks!

Answer 1

First, in the theorem, you have to assume that manifolds are connected.

Next, it is a theorem (Rado and Caratheodory, I think), that every topological surface has a smooth structure and that two surfaces are homeomorphic if and only if they are diffeomorphic.

See here for more references than you probably wanted.

As for Riemann surfaces: if you want to classify them up to biholomorphic maps, then things become complicated. Restrict to genus $g\ge 2$ compact surfaces. Then the space of complex structures is a complex manifold of dimension $3g-3$, called the moduli space. For $g=1$, the moduli space is 1-dimensional, while for the sphere (genus zero), the complex structure is unique.