Riemann surface: Difference between revisions

Revision as of 22:29, 3 May 2008

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.

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