---
tag: PZZ4X908Y8
kind: definition
name: Lagrangian submanifold
msc: [53D12]
defines: H8ACPTMX08
uses: [CAN490EPRT, A8TS2SNX9Z, 0E3G0GRPZ6, 0ZV14XRNF0]
sources:
  - url: https://en.wikipedia.org/wiki/Symplectic_manifold
---

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$.
Let $L \subseteq M$ be a submanifold with inclusion map $\iota \colon L \to M$.
The submanifold $L$ is *Lagrangian* if:

1. $\dim L = n$, and
2. $\iota^* \omega = 0$, that is $\omega_p(v, w) = 0$ for all $p \in L$ and $v, w \in T_p L$.

## Remarks

Example: in $\R^4$ with $\omega_0 = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$, the plane $L = \set{y_1 = y_2 = 0}$.
It has dimension $2$, and on it $dy_1 = dy_2 = 0$, so $\iota^* \omega_0 = 0$.

Non-example: the plane $P = \set{x_2 = y_2 = 0}$ in the same space has dimension $2$, but $\omega_0$ restricts to $dx_1 \wedge dy_1 \ne 0$ on it, so clause 2 fails.
Such a plane is called symplectic.

Clause 2 alone forces $\dim L \le n$, so Lagrangian submanifolds are the isotropic submanifolds of largest dimension.
In a cotangent bundle $T^*Q$ with its canonical form, the graph of a $1$-form $\alpha$ on $Q$ is Lagrangian exactly when $\alpha$ is [[closed|0ZV14XRNF0]].
A diffeomorphism $\phi \colon M \to M$ is a [[symplectomorphism|A8TS2SNX9Z]] exactly when its graph is Lagrangian in $M \times M$ with the form $\mathrm{pr}_1^* \omega - \mathrm{pr}_2^* \omega$.