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Further definitions and properties

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The statement of the theorem

As with any map between complex manifolds, a functionf : M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h ∘ f ∘ g^{−1} 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 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. Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f(g^{−1}(z)), h can be considered as a map from an open set of R^{2} to R^{2} whose Jacobian in a point z is just the real linear map given by multiplication by the complex numberh′(z). However, the real determinant of multiplication by a complex number α equals |α|^{2}, so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas. 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 functions (holomorphic functions with values in the Riemann sphereC ∪ {∞}). More precisely, the function field of X is a finite 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 Siegel (1955). Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map of the surface. All compact Riemann surfaces are algebraic curves since they can be embedded into some CP^{n}. This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve. - ^Nollet, Scott. "KODAIRA'S THEOREM AND COMPACTIFICATION OF MUMFORD'S MODULI SPACE Mg"(PDF).