---
tag: Q2DMSTWZYZ
kind: concept
name: Legendrian submanifold
msc: [53D10, 53D12, 57K33]
sources:
  - url: https://en.wikipedia.org/wiki/Contact_geometry
---

## Summary

A Legendrian submanifold of a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$ is a submanifold of dimension $n$ that is everywhere tangent to the contact hyperplanes.
Legendrian submanifolds play in contact geometry the role that [[Lagrangian submanifolds|H8ACPTMX08]] play in symplectic geometry.
In a contact $3$-manifold they are Legendrian knots, whose classification up to Legendrian isotopy is a rich subject with invariants such as the Thurston–Bennequin number.

## Explanations

### L1 human

A contact structure gives a plane at each point of a three-dimensional space, and these planes twist so much that no surface can follow them.
Curves can still follow them: a Legendrian curve is a curve whose velocity always lies in the given plane.
In $\R^3$ with the planes $\ker(dz - y\,dx)$, the curve $t \mapsto (t, f'(t), f(t))$ is Legendrian for every smooth function $f$, because $dz - y\,dx$ vanishes on its velocity.
Legendrian knots appear in geometric optics as the fronts of propagating waves, lifted with their direction of travel.