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  "body": "\n## Definition\n\nLet $(E, \\xi)$ be a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$ and let $\\rho \\colon E \\to B$ be a smooth fibre bundle over a [[smooth manifold|F45VQ47GPV]] $B$ of dimension $n + 1$.\nThe bundle $\\rho$ is a *Legendrian fibration* if each fibre is a [[Legendrian submanifold|Q2DMSTWZYZ]] of $E$.\nLet $\\iota \\colon L \\to E$ be a Legendrian [[immersion|6RW68JGBHC]], that is, $\\dim L = n$ and $d\\iota$ maps each tangent space of $L$ into $\\xi$.\nThe composition $\\rho \\circ \\iota \\colon L \\to B$ is a *Legendrian map*.\nIts *wave front* (or *front*) is the image $\\rho(\\iota(L)) \\subset B$.\n\n## Remarks\n\nThe model case is the [[contact structure on the jet space|GCS702QT54]] $J^1(Q) = T^*Q \\times \\R$ of a manifold $Q$ of dimension $n$, with coordinates $(q, p, z)$, contact form $dz - \\sum_i p_i \\, dq_i$, and $\\rho(q, p, z) = (q, z) \\in Q \\times \\R$.\nThe fibres $\\set{(q, z)} \\times T_q^* Q$ are Legendrian, since $dq = 0$ and $dz = 0$ along them.\n\nExample: $Q = \\R$ and $\\iota(t) = (t^2, t, \\tfrac23 t^3)$.\nAlong $\\iota$, $dz = 2t^2 \\, dt$ and $p \\, dq = t \\cdot 2t \\, dt = 2t^2 \\, dt$, so $\\iota^*(dz - p\\,dq) = 0$ and $\\iota$ is a Legendrian embedding.\nIts 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.\n\nNon-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.\n\nFor the $1$-jet lift $q \\mapsto (q, df(q), f(q))$ of a smooth function $f$, the front is the graph of $f$.\nWhere the front is a smooth hypersurface not tangent to the $z$-direction, the Legendrian submanifold is recovered from it by $p = \\partial z / \\partial q$.\n",
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