a Riemann surface
/ REE-mahn /
Imagine you want to do complex calculus — differentiate functions of z = x + iy — but not just on the flat plane. You want to do it on a curved shape: a sphere, a doughnut, a pretzel with several holes. A Riemann surface is exactly such a shape, equipped with enough structure that the phrase 'holomorphic function near a point' makes sense everywhere on it. Locally it looks like an open piece of the complex plane; globally it can wrap around and close up.
Precisely, a Riemann surface is a connected complex manifold of complex dimension one. It is a Hausdorff topological space covered by charts to open sets of C, where every transition map (going from one chart's coordinate to another's, on the overlap) is holomorphic — not merely smooth, but complex-analytic. Because a holomorphic map of one complex variable is conformal where its derivative is nonzero, this is the same as carrying a conformal structure: an unambiguous notion of angle but not of length. As a real object every Riemann surface is a smooth orientable real surface of dimension two; the extra demand is that the gluing be by holomorphic, not just smooth, maps.
Why it matters: this is where complex analysis, topology, and algebra fuse. The familiar examples — the Riemann sphere C union {infinity} (the genus-0 case), a complex torus C/L for a lattice L (genus 1), and the smooth zero set of a polynomial like y^2 = x^5 - 1 (higher genus) — are the basic objects of the whole subject. A key honesty point: 'Riemann surface' is a complex-analytic notion, strictly richer than the underlying real surface. Two tori C/L and C/L' are the same real torus topologically but generally NOT isomorphic as Riemann surfaces; the conformal structure carries genuine extra information (their modulus). The word 'surface' refers to real dimension two, even though complex dimension is one.
The Riemann sphere is built from two charts: U_0 = C with coordinate z, and U_1 = C with coordinate w, glued over the overlap by the holomorphic map w = 1/z. This single rule turns two copies of the plane into a sphere on which 'infinity' is just an ordinary point (w = 0), and a rational function like 1/z becomes a perfectly nice holomorphic map to the sphere.
Two planes glued by w = 1/z become the sphere; infinity is an ordinary point in the second chart.
A Riemann surface is more than its underlying real surface: the conformal (holomorphic) structure is extra data. All genus-1 surfaces are the same topological torus, but they form a one-parameter family of pairwise-inequivalent Riemann surfaces — never conflate the two.