---
tag: H8ACPTMX08
kind: concept
name: Lagrangian submanifold
msc: [53D12]
sources:
  - url: https://en.wikipedia.org/wiki/Symplectic_manifold
---

## Summary

A Lagrangian submanifold of a symplectic manifold is a submanifold of half the dimension on which the symplectic form vanishes.
Lagrangian submanifolds are the largest submanifolds on which the form vanishes identically.
They include the zero section and the graphs of closed $1$-forms in cotangent bundles, and the graphs of [[symplectomorphisms|A8TS2SNX9Z]].
They are central objects of symplectic topology, Floer theory and mirror symmetry.

## Explanations

### L1 human

A symplectic form measures areas of small surfaces inside a manifold of even dimension $2n$.
A Lagrangian submanifold has dimension $n$ and is so placed that every small surface inside it has zero symplectic area.
In the phase plane with position $q$ and momentum $p$, every curve is Lagrangian.
In $\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, the plane where both momenta are zero is Lagrangian.