Wave front of a Legendrian map

Remarks

The model case is the contact structure on the jet space $J^1(Q) = T^*Q \times \R$ of a manifold $Q$ of dimension $n$, with coordinates $(q, p, z)$, contact form $dz - \sum_i p_i \, dq_i$, and $\rho(q, p, z) = (q, z) \in Q \times \R$. The fibres $\set{(q, z)} \times T_q^* Q$ are Legendrian, since $dq = 0$ and $dz = 0$ along them.

Example: $Q = \R$ and $\iota(t) = (t^2, t, \tfrac23 t^3)$. Along $\iota$, $dz = 2t^2 \, dt$ and $p \, dq = t \cdot 2t \, dt = 2t^2 \, dt$, so $\iota^*(dz - p\,dq) = 0$ and $\iota$ is a Legendrian embedding. Its front is the curve $(t^2, \tfrac23 t^3)$ in the $(q, z)$-plane, a semicubical parabola with a cusp at the origin, although $\iota$ itself is smooth with nonzero velocity.

Non-example: the curve $\gamma(t) = (t, 0, t)$ in $J^1(\R)$ is not Legendrian, since $\gamma^*(dz - p\,dq) = dt \ne 0$; so its projection $(t, t)$ is not a front in this sense.

For the $1$-jet lift $q \mapsto (q, df(q), f(q))$ of a smooth function $f$, the front is the graph of $f$. Where the front is a smooth hypersurface not tangent to the $z$-direction, the Legendrian submanifold is recovered from it by $p = \partial z / \partial q$.