Riemann surfaces are complex planes with wild global shapes
Named for Bernhard Riemann, these objects are connected one-complex-dimensional manifolds—locally like open disks in the complex plane, yet globally able to look like spheres, tori, or glued sheets. They carry extra complex or conformal structure beyond an ordinary two-sided surface.
Complex analysts treat a Riemann surface as a linked, one-complex-dimensional manifold. Locally it looks like an open disk in the plane of complex numbers, yet globally it may mimic a sphere, a torus, or several sheets glued along cuts. Multivalued function graphs are textbook models. Viewed with real coordinates the object is an ordinary two-dimensional surface that also carries complex data. Only orientable, metrizable real surfaces can receive that extra structure—spheres and tori qualify, while Möbius strips, Klein bottles, and the real projective plane do not.
Several equivalent packages encode the same idea. One starts with a connected Hausdorff space and an atlas of maps into the open unit disk whose overlap changes are holomorphic. Another begins with a two-sided real surface plus a conformal structure: a class of Riemannian metrics that agree on angles. Pushing the plane's Euclidean metric through the charts turns a complex atlas into conformal data; going the other way is subtler. Holomorphic maps compose. When a holomorphic bijection with holomorphic inverse links two surfaces they are biholomorphic—also called conformally equivalent—and behave as one object for applications.
Because multiplying by a nonzero complex scalar has positive real determinant, a complex atlas automatically orients the underlying real surface. Away from compactness one always finds non-constant holomorphic maps into the finite complex plane, and such surfaces are Stein. Compactness flips the story: the maximum principle locks every finite-valued holomorphic map to a constant, yet meromorphic maps into the extended sphere still vary. Any two of those meromorphic maps are algebraically dependent, so the field they generate is that of a curve in one variable.
Those meromorphic maps, together with compactness, force algebraicity: the surface embeds in projective space—even complex projective three-space—cut out by polynomials. Gluing charts locally plus demanding compactness globally therefore produces an algebraic curve, a rigidity that surprises newcomers. Curvature sorts surfaces into hyperbolic, parabolic, and elliptic classes with negative, zero, or positive constant sectional curvature. Each connected example admits a unique complete constant-curvature metric inside its conformal class, a fact tied to isothermal coordinates.
Source: Riemann surface
