Lagrangian submanifold

拉格朗日子流形

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

Translation evidence: Literature usage

Meaning in context

An embedded smooth submanifold of a symplectic manifold of dimension 2n whose dimension is n and on whose tangent spaces the symplectic form vanishes. The submanifold need not be a closed subset under this entry's convention.

在 $2n$ 维辛流形中的 $n$ 维光滑嵌入子流形，其各切空间上辛形式均为零；本条不要求子流形在环境空间中为闭集。

Example

For $\omega=\sum dq_i\wedge dp_i$ on $T^*\mathbb R^n$, the graph $p_i=\partial f/\partial q_i$ is Lagrangian: the symmetric Hessian cancels in the pulled-back two-form.

Usage and distinctions

Vanishing of the restricted form alone gives isotropy; the half-dimension condition is essential. Cannas da Silva uses a closed-embedding convention that is relaxed explicitly here.

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Definition sources

1. Ana Cannas da Silva — Lectures on Symplectic Geometry§3.1, Definition 3.2; §3.2, Definition 3.3 and Proposition 3.4 · Checked 2026-10-14

Translation sources

1. Wang Yi — Beijing Institute of Technology, 弦拓扑和拉格朗日子流形 report noticeReport title 弦拓扑和拉格朗日子流形 and abstract containing 拉格朗日子流形（Lagrangian submanifold）; PDF p. 1 · Checked 2026-10-14

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