Further information: Complex manifold and Conformal geometry There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. A Riemann surface is an 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.

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Further information: Complex manifold and Conformal geometry There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. A Riemann surface is an 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 which 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.

A torus. The complex plane C is the most basic Riemann surface. The charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In an analogous fashion, every non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every non-empty open subset of a Riemann surface is a Riemann surface.

This particular surface is called the Riemann sphere because it can be interpreted as wrapping the complex plane around the sphere. Unlike the complex plane, it is compact. The theory of compact Riemann surfaces can be shown to be equivalent to that of projective algebraic curves that are defined over the complex numbers and non-singular. This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve.

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## Riemann surface

Moogulrajas Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces Evidently, this is equivalent to saying that the family of complements contains no finite covering. It has been found most convenient to base the definition on the consideration of open coverings. As a consequence of 3B every point in aglfors topological space belongs to a maximal connected subset, namely the union of all connected sets which contnin the given point. If 01 is not empty it meets at least one P.

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## Lars Valerian Ahlfors

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## Ahlfors theory

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