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Riemann surface

Not to be confused with Riemannian surface or Riemannian manifold.
For the Riemann surface of a subring of a field, see Zariski–Riemann space.
The Riemann surface for the multivalued complex function f(z) = log(z) in a neighborhood of the origin. The (x, y) coordinates are the coordinates of z in the complex plane; the vertical coordinate represents the imaginary part of f(z), whose real part is represented by hue.

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

Examples of Riemann surfaces include graphs of multivalued functions such as √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. Given this, 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.

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