symplectic manifold

辛流形

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Translation evidence: Literature usage

Meaning in context

A smooth manifold equipped with a smooth closed two-form whose value on every tangent space is nondegenerate; closed means its exterior derivative vanishes, and nondegenerate means contraction identifies tangent and cotangent spaces.

配有光滑闭二形式的光滑流形，且该形式在各切空间上非退化；闭指外微分为零，非退化指缩并给出切空间到余切空间的同构。

Example

On $\mathbb R^{2n}$, $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ is closed and has invertible skew matrix, so it defines a symplectic manifold.

Usage and distinctions

Even dimension is necessary but does not itself provide a symplectic form. Unlike a Riemannian metric, the form is alternating and gives no lengths of individual vectors.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Ana Cannas da Silva — Lectures on Symplectic Geometry§1.3, Definitions 1.5–1.6 and the standard Euclidean example · Checked 2026-10-14

Translation sources

1. Jinan University — Wang Bai-Ling, Lectures on fundamental symplectic geometry I内容简介：辛流形及其子流形；数学系学术讲座(十六), published 2020-12-14 · Checked 2026-10-14

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