---
tag: 824SSQ3SQN
kind: concept
name: Wave front of a Legendrian map
msc: [53D10, 58K05]
sources:
  - url: https://en.wikipedia.org/wiki/Contact_geometry
---

## Summary

Let a [[contact manifold|HBQ8BZ7XB5]] be fibred over a base by a Legendrian fibration.
The wave front of a Legendrian map is the image of a [[Legendrian submanifold|Q2DMSTWZYZ]] in this base.
In the model case of the space of $1$-jets $J^1(Q) = T^*Q \times \R$, it is the projection of the Legendrian submanifold to $Q \times \R$.
The front is usually a hypersurface with singularities such as cusps.
At its smooth points the front determines the Legendrian submanifold through its tangent hyperplanes.
Wave fronts are the contact counterpart of [[caustics|GMRYG4S3G0]], and they describe the moving fronts of waves in geometric optics.

## Explanations

### L1 human

Drop a stone in a pond with a curved bank: the ripple reflected from the bank is a curve that moves and changes shape.
At some moments this curve develops sharp corners called cusps.
The ripple is a wave front, and it is the shadow of a smooth curve one dimension up.
That smooth object records, at each point of the front, also the direction in which the front is facing.
For a function $f$ of one variable, the front of its $1$-jet curve $(q, f'(q), f(q))$ is just the graph of $f$.
The semicubical cusp $(t^2, \tfrac23 t^3)$ is a front that is not a graph near $0$.