Definitions
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The statement of the theorem
There are several equivalent definitions of a Riemann surface.
- A Riemann surface X is a connectedcomplex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and the transition maps between two overlapping charts are required to be holomorphic.
- A Riemann surface is a (connected) oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that X is endowed with an additional structure that allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.
A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to X by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.
- ^Farkas & Kra 1980, Miranda 1995
- ^See (Jost 2006, Ch. 3.11) for the construction of a corresponding complex structure.