Riemann surface - Knowlpedia

A Riemann surface is a connected complex manifold of complex dimension $1$ . Explicitly, it is a connected Hausdorff, second-countable space $X$ with an atlas of homeomorphisms $z_\alpha:U_\alpha\to V_\alpha\subseteq\mathbb C$ whose transition maps $z_\beta\circ z_\alpha^{-1}$ are holomorphic wherever defined. Thus $X$ is an underlying real smooth manifold of dimension $2$ , while its complex charts specify which local complex-valued functions and maps are holomorphic.

Holomorphic structure

A function $f:X\to\mathbb C$ is holomorphic precisely when $f\circ z_\alpha^{-1}$ is holomorphic in one complex variable in every chart. These functions form the sheaf of holomorphic functions $\mathcal O_X$ . A map between Riemann surfaces is holomorphic when its coordinate expressions are holomorphic. Every bijective holomorphic map between Riemann surfaces is automatically biholomorphic: injectivity rules out a zero local derivative, so the holomorphic inverse function theorem applies in charts.

Underlying orientation

Every Riemann surface has a canonical orientation as a real surface. Indeed, a holomorphic transition map with nonzero complex derivative has positive real Jacobian determinant $\lvert f'(z)\rvert^2$ , so its complex charts are orientation-compatible. This orientation is part of the structure induced by the complex atlas, not extra data chosen afterward.

Examples and conventions

Connected open subsets of $\mathbb C$ , the Riemann sphere $\mathbb P^1(\mathbb C)$ , complex tori $\mathbb C/\Lambda$ , and connected nonsingular complex plane curves are standard examples. More generally, each connected component of a nonsingular complex curve is a Riemann surface. A topological surface alone is not a Riemann surface until a complex structure is specified. Some authors allow a Riemann surface to be disconnected; here connectedness is part of the definition. “Complex curve” may also mean a singular analytic space or an algebraic curve, whereas this knowl uses it only for a nonsingular one-dimensional complex manifold.

References

Otto Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer, 1981. Publisher record. Relevant: Chapter 1, §§1–2 and §6.