Riemann surface

黎曼曲面

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A connected Hausdorff, second countable topological surface without boundary equipped with an atlas to open subsets of the complex plane whose transition maps are holomorphic; equivalently, a connected complex manifold of complex dimension one.

具有到复平面开集的坐标图册、且坐标变换全纯的连通、Hausdorff、第二可数无边拓扑曲面，即复维数为一的连通复流形。

Example

The sphere $\mathbb C\cup\{\infty\}$ has compatible coordinates $z$ away from infinity and $w=1/z$ near infinity.

Usage and distinctions

A Riemann surface has complex coordinate structure; a Riemannian surface has a metric. This entry requires connectedness and excludes boundary points, while some authors use a disconnected convention.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Constantin Teleman — Riemann SurfacesLecture 3, Definition 3.1 and Proposition 3.2, printed p. 10; connected-component convention discussed after Theorem 3.5 · Checked 2026-10-15

Translation sources

1. University of Chinese Academy of Sciences — Deng Fusheng, course syllabus第一章 黎曼曲面与复流形的基本概论，第2节：黎曼曲面与复流形的定义 · Checked 2026-10-15

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