Riemann surface: Difference between revisions

Revision as of 01:15, 1 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 $U_{α}$ , and homeomorphisms $φ_{α} : U_{α} \to V_{α}$ where $V_{α}$ are open subsets of $C$ , such that the transition maps $φ_{β} \circ φ_{α}^{- 1} : φ_{α} (U_{α} \cap U_{β}) \to φ_{β} (U_{α} \cap U_{β})$ , are all biholomorphic mappings.

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	==Definition==		==Definition==

	A '''Riemann surface''' is a ~~two~~-dimensional [[complex manifold]]: it is a 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.