Editing Riemann surface (section)

== Further definitions and properties ==
As with any map between complex manifolds, a [[function (mathematics)|function]] {{nowrap|''f'' : ''M'' → ''N''}} between two Riemann surfaces ''M'' and ''N'' is called ''[[Holomorphic function|holomorphic]]'' if for every chart ''g'' in the [[atlas (topology)|atlas]] of ''M'' and every chart ''h'' in the atlas of ''N'', the map {{nowrap|''h'' ∘ ''f'' ∘ ''g''<sup>−1</sup>}} is holomorphic (as a function from '''C''' to '''C''') wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces ''M'' and ''N'' are called ''[[Biholomorphism|biholomorphic]]'' (or ''conformally equivalent'' to emphasize the conformal point of view) if there exists a [[bijective]] holomorphic function from ''M'' to ''N'' whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent Riemann surfaces are for all practical purposes identical.

=== Orientability ===
Each Riemann surface, being a complex manifold, is [[orientable]] as a real manifold. For complex charts ''f'' and ''g'' with transition function {{nowrap|1=''h'' = ''f''(''g''<sup>−1</sup>(''z''))}}, ''h'' can be considered as a map from an open set of '''R'''<sup>2</sup> to '''R'''<sup>2</sup> whose [[Jacobian matrix and determinant|Jacobian]] in a point ''z'' is just the real linear map given by multiplication by the [[complex number]] ''h''′(''z''). However, the real [[determinant]] of multiplication by a complex number ''α'' equals {{abs|''α''}}<sup>2</sup>, so the Jacobian of ''h'' has positive determinant. Consequently, the complex atlas is an oriented atlas.

=== Functions ===
Every non-compact Riemann surface admits non-constant holomorphic functions (with values in '''C'''). In fact, every non-compact Riemann surface is a [[Stein manifold]].

In contrast, on a compact Riemann surface ''X'' every holomorphic function with values in '''C''' is constant due to the [[maximum principle]]. However, there always exist non-constant [[meromorphic function]]s (holomorphic functions with values in the [[Riemann sphere]] {{nowrap|'''C''' ∪ {{mset|∞}}}}). More precisely, the [[function field of an algebraic variety|function field]] of ''X'' is a finite [[field extension|extension]] of '''C'''(''t''), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see {{harvtxt|Siegel|1955}}. Meromorphic functions can be given fairly explicitly, in terms of Riemann [[theta function]]s and the [[Abel–Jacobi map]] of the surface.

=== Algebraicity ===

All compact Riemann surfaces are [[algebraic curves]] since they can be embedded into some '''CP'''<sup>''n''</sup>. This follows from the [[Kodaira embedding theorem]] and the fact there exists a positive line bundle on any complex curve.<ref>{{cite web|url=http://faculty.tcu.edu/richardson/Seminars/scott4-15.pdf|title=KODAIRA'S THEOREM AND COMPACTIFICATION OF MUMFORD'S MODULI SPACE Mg|last=Nollet|first=Scott}}</ref>