Wave front of a Legendrian map

Summary

Let a contact manifold be fibred over a base by a Legendrian fibration. The wave front of a Legendrian map is the image of a Legendrian submanifold 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, and they describe the moving fronts of waves in geometric optics.