Riemann Surface

A Riemann surface is a connected one-dimensional complex manifold, first studied by and 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 their global topology can differ greatly, resembling a sphere, a torus, or several sheets glued together, as seen in the graphs of multivalued functions such as the square root or the logarithm. Every Riemann surface is a surface in the sense of a two-dimensional real manifold, but carries the additional structure of a complex structure, and 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; under this correspondence the sphere and torus admit complex structures but the Mobius strip, Klein bottle and real projective plane do not. Every compact Riemann surface is a complex algebraic curve, by the Riemann-Roch theorem and Chow's theorem.