Noun
Riemann surface (plural Riemann surfaces)
(mathematics) A one-dimensional complex manifold; a generalization of the complex plane.
Any complex nonsingular algebraic curve viewed as a complex manifold is a Riemann surface. Source: Internet
For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface. Source: Internet
For the Riemann surface has parameters (the "moduli"). Source: Internet
Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by C(M). Source: Internet
Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented as a torus. Source: Internet