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 ${\phi }_{\alpha }:U_{\alpha }\to V_{\alpha }$ where $V_{\alpha }$ are open subsets of $\mathbb{C}$, such that the transition maps ${\phi }_{\beta }\circ \underset{\alpha }{\overset{-1}{\phi }}:{\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\beta }(U_{\alpha }\cap U_{\beta })$, 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.