almost complex structure

近复结构

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

Translation evidence: Literature usage

Meaning in context

A smooth real bundle endomorphism of the tangent bundle of a smooth manifold whose square is minus the identity on every tangent space.

光滑流形切丛的光滑实丛自同态，在每个切空间上其平方等于负恒等映射。

Example

On $\mathbb R^2$, set $J\partial_x=\partial_y$ and $J\partial_y=-\partial_x$. Then $J^2=-\operatorname{id}$, the usual complex structure.

Usage and distinctions

An almost complex structure is pointwise data and need not arise from holomorphic charts. Compatibility with a specified symplectic form is a separate positive-definiteness condition.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Ana Cannas da Silva — Lectures on Symplectic Geometry§12.3, Definitions 12.4–12.5 · Checked 2026-10-14

Translation sources

1. Li Xiaobin — Southwest Jiaotong University, Gromov–Witten study guide§2.2 辛几何，核心内容：近复结构 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.