Riemann surface - Imported Knowlpedia Content

Definition

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; a bijective holomorphic map has holomorphic inverse when it is locally biholomorphic.

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. The basic manifold and mapping conventions are presented in Forster, §§1–2.

Examples and conventions

Open subsets of $\mathbb C$ , the Riemann sphere $\mathbb P^1(\mathbb C)$ , complex tori $\mathbb C/\Lambda$ , and nonsingular complex plane curves are standard examples. 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.