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Riemann surface
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Diagram showing, at left, pieces of a mathematical surface defined by f(x, y) = 0 and their projection onto the x-plane of complex numbers. At right, the surface is cut along labeled curves to form an octagonal shape, with numbered and lettered edges indicating how the surface is unfolded in topology and Riemannian geometry.
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Riemann surface

mathematics
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Diagram showing, at left, pieces of a mathematical surface defined by f(x, y) = 0 and their projection onto the x-plane of complex numbers. At right, the surface is cut along labeled curves to form an octagonal shape, with numbered and lettered edges indicating how the surface is unfolded in topology and Riemannian geometry.
cutting a Riemann surface (Left) Pieces of a surface given by f(x, y) = 0; (right) if the surface is cut along the curves, an octagon is obtained.

A Riemann surface is a multilayered surface used in complex function theory. It allows a multivalued complex function to be treated as a single-valued function of a complex variable.1

This concept was introduced by Bernhard Riemann in his doctoral thesis in 1851. He generalized the study of polynomial equations in two real variables to two complex variables. In this context, an equation involving two complex variables defines a real surface spread out over the plane, now known as a Riemann surface. Riemann showed how these surfaces could be classified by a number called the genus, which is determined by the maximum number of closed curves that can be drawn on the surface without splitting it into separate pieces. This was an early significant use of topology in mathematics.2

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  1. Riemann surface
  2. Bernhard Riemann

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