Lectures on String Theory

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coordinate chart
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Fig. 6. Covering the Riemann surface with coordinate patches. Every patch
is homeomorphic to an open domain of the Euclidean plane.
Second, in order to prove that isothermal coordinates exist, consider the so-called
Beltrami equation
∂w
= µ(z, ¯z)∂w
∂¯z ∂z .
Suppose we solve this equation, then
Thus,
|dw| 2 = |∂ z w + ∂¯z w| 2 = |∂ z w| 2 |dz + µd¯z| 2 = |∂ zw| 2
2e φ ds2 .
ds 2 =
2eφ
|∂ z w| 2 |dw|2 ≡ λ|dw| 2 ,
i.e. w defines a system of isothermal coordinates. It is a mathematical theorem that
for a sufficiently small coordinate patch and a differentiable metric a solution of the
Beltrami equation with ∂ z w ≠ 0 always exists.
If two metrics are related by a diffeomorphism and a Weyl rescaling they are said
to define the same conformal structure. If a manifold M is covered by a system of
conformal (isothermal) coordinate patches U α , then on the overlaps the metrics are
conformally related, i.e. the transition functions on the overlaps U α ∩U β are analytic
and the complex coordinates are globally defined. A system of analytic coordinate
patches is called a complex structure and it is the same as a conformal structure.
A two-dimensional topological manifold endowed with a complex structure is
called a Riemann surface. Thus, a Riemann surface is a complex manifold.
Another way to understand that Riemann surface is a complex manifold is to
note that in two dimensions the metric provides a globally-defined integrable complex
structure
I α β = √ hh αγ ɛ γβ
such that I 2 = −1. The conformal structure is conformally-invariant and globally
well-defined. The existence of an integrable complex structure is necessary and sufficient
for an even-dimensional manifold to be complex.