symplectic form

辛形式

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

Translation evidence: Literature usage

Meaning in context

A smooth two-form on a smooth manifold whose exterior derivative is zero and whose contraction map from each tangent space to its dual is invertible.

光滑流形上的光滑二形式，外微分为零，且在每一点由缩并得到的切空间到其对偶空间的映射均可逆。

Example

The form $dx\wedge dy$ on $\mathbb R^2$ is symplectic. The form $x\,dx\wedge dy$ is closed but degenerates where $x=0$.

Usage and distinctions

An alternating form can be nondegenerate pointwise and still fail closedness in dimension at least four. A symplectic form is the additional structure; the pair with its manifold is a symplectic manifold.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Ana Cannas da Silva — Lectures on Symplectic Geometry§1.3, Definition 1.5; §1.1, nondegenerate alternating forms · 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：辛形式 · Checked 2026-10-14

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