Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

Not to be confused with Riemannian surface or Riemannian manifold.
For the Riemann surface of a subring of a field, see Zariski–Riemann space.

Loosely speaking, this means that any Riemann surface is formed by gluing together open subsets of the complex plane C using holomorphic gluing maps.

Examples of Riemann surfaces include graphs of multivalued functions like √z or log(z), e.g. the subset of pairs (z,w) ∈ C2 with w = log(z).

Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure (specifically a complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not.

Every compact Riemann surface is a complex algebraic curve by Chow's theorem and the Riemann–Roch theorem.

Riemann surfaces were first studied by and are named after Bernhard Riemann.

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