pseudoholomorphic curve

伪全纯曲线

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

Translation evidence: Literature usage

Meaning in context

A smooth map from a Riemann surface with complex structure j to a smooth manifold with almost complex structure J whose differential intertwines those structures, so du composed with j equals J composed with du.

从复结构为 $j$ 的黎曼曲面到近复结构为 $J$ 的光滑流形的光滑映射 $u$，满足 $du\circ j=J\circ du$。

Example

For the usual structures on $\mathbb C$, $u(z)=z^2$ is pseudoholomorphic; $u(z)=\overline z$ is not, since its differential anticommutes with multiplication by $i$.

Usage and distinctions

The word curve denotes a parametrized map, not just its image. The map need not be embedded or immersive; branch points and constant maps are allowed by this definition.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Chris Wendl — Lectures on Holomorphic Curves in Symplectic and Contact Geometry§2.1, Definition 2.8 and equation (2.2), printed p. 26 · Checked 2026-10-14

Translation sources

1. Jinan University — Wang Bai-Ling, Lectures on fundamental symplectic geometry I内容简介：伪全纯曲线；数学系学术讲座(十六), published 2020-12-14 · Checked 2026-10-14

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