Riemann surface

Definition

A Riemann surface is a connected one-dimensional complex manifold: it is a connected second-countable Hausdorff space $M$ equipped with an atlas of coordinate charts with all the transition maps being biholomorphic. More explicitly, it is a second-countable Hausdorff space $M$ along with an open cover $U_{\alpha }$, and homeomorphisms $\varphi _{\alpha }:U_{\alpha }\to V_{\alpha }$ where $V_{\alpha }$ are open subsets of $\mathbb {C}$, such that the transition maps $\varphi _{\beta }\circ \varphi _{\alpha }^{-1}:\varphi _{\alpha }(U_{\alpha }\cap U_{\beta })\to \varphi _{\beta }(U_{\alpha }\cap U_{\beta })$, are all biholomorphic mappings.

Note that since conformal maps are in particular orientation-preserving, any Riemann surface is orientable; in fact, the conformal structure prescribes an orientation to the Riemann surface.