Lectures on String Theory

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5.2 Riemann surfaces
On a two-dimensional real manifold M the metric can be locally (i.e. in a given
coordinate chart) defined by the line element
Introducing the complex coordinates
ds 2 = h 11 dx 2 + 2h 12 dxdy + h 22 dy 2
z = x + iy ,
¯z = x − iy
the line element can be written as
ds 2 = 2e φ |dz + µd¯z| 2
with the following identifications
(
√
)
e φ = 1 h
8 11 + h 22 + 2 h 11 h 22 − h 2 h 11 − h 22 + 2ih 12
12 , µ =
h 11 + h 22 + 2 √ .
h 11 h 22 − h 2 12
If h 11 = h 12 and h 12 = 0 then µ = 0 and the metric takes in this coordinate chart
the form
ds 2 = 2e φ |dz| 2 = 2e φ (dx 2 + dy 2 ) .
The corresponding coordinate system is called isothermal or conformal and the coordinates
(x, y) define a conformal map of a coordinate chart of a manifold to the
Euclidean plane.
A theorem of Gauss
For any real two-dimensional orientable surface with a positive definite metric there
always exists a system of isothermal coordinates (the theorem of Gauss). It is unique
up to conformal transformations. First, assume that we have already found a system
of isothermal coordinates, i.e. the metric is locally in the form
ds 2 = 2e φ |dz| 2 .
Performing the coordinate transformation with an analytic function of z:
we get
z → f(z)
ds 2 → ds 2 = 2e φ |f(z)| 2 |dz| 2 ,
i.e. we get a conformally equivalent metric and, therefore, a new system of the
isothermal coordinates.