Editing Riemann surface (section)

== Definitions ==
{{further|Complex manifold|Conformal geometry}}
There are several equivalent definitions of a Riemann surface.

# A Riemann surface ''X'' is a [[Connected space|connected]] [[complex manifold]] of [[complex dimension]] one. This means that ''X'' is a connected [[Hausdorff space]] that is endowed with an [[atlas (topology)|atlas]] of [[chart (topology)|chart]]s to the [[open unit disk]] of the [[complex plane]]: for every point {{nowrap|''x'' ∈ ''X''}} there is a [[neighbourhood (topology)|neighbourhood]] of ''x'' that is [[homeomorphic]] to the open unit disk of the complex plane, and the [[transition map]]s between two overlapping charts are required to be [[Holomorphic function|holomorphic]].<ref>{{harvnb|Farkas|Kra|1980}}, {{harvnb|Miranda|1995}}</ref>
# A Riemann surface is a (connected) [[oriented manifold]] of (real) dimension two – a two-sided [[Surface (topology)|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 metric]]s. Two such metrics are considered [[conformally equivalent|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.<ref>See {{Harvard citations|author=Jost|year=2006|loc=Ch. 3.11}} for the construction of a corresponding complex structure.</ref>