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  "body": "\n## Definition\n\nLet $\\pi \\colon E \\to M$ be a smooth fiber bundle, let $k \\ge 1$, and let $J^k\\pi$ be its $k$-th [[jet bundle|HC3VBBCJQP]].\nA [[one-form|02Z5B2HA3F]] $\\theta$ on an open set of $J^k\\pi$ is a *contact form* if\n$$(j^k s)^* \\theta = 0$$\nfor every local section $s$ of $\\pi$.\nHere $j^k s$ is the [[prolongation|BMCCQ8JXNE]] of $s$, and the pullback is taken where it is defined.\nThe *contact distribution*, or Cartan distribution, of $J^k\\pi$ is the subbundle $\\mathcal{C} \\subseteq T J^k\\pi$ of vectors on which every contact form vanishes.\nThe contact forms, or equivalently the distribution $\\mathcal{C}$, form the *contact structure* of $J^k\\pi$.\n\n## Remarks\n\nIn adapted coordinates $(x^i, u^\\alpha, u^\\alpha_I)$, the contact forms are the combinations with smooth coefficients of\n$$\\theta^\\alpha_I = du^\\alpha_I - \\sum_{i} u^\\alpha_{I + i}\\, dx^i, \\qquad 0 \\le \\abs{I} \\le k - 1,$$\nwhere $u^\\alpha_\\emptyset = u^\\alpha$ and $I + i$ is the multi-index $I$ with $i$ added once.\n\nA local section $\\sigma$ of $J^k\\pi \\to M$ is holonomic, that is, of the form $j^k s$, exactly when $\\sigma^*\\theta = 0$ for every contact form $\\theta$.\nFor $k = 1$, $M = \\R$ and one fiber coordinate this is a direct check: if $\\sigma(x) = (x, s(x), q(x))$ then $\\sigma^*(du - q\\, dx) = (s'(x) - q(x))\\, dx$, which vanishes exactly when $q = s'$.\n\nExample: for $\\pi \\colon \\R \\times \\R \\to \\R$, $J^1\\pi = \\R^3$ with coordinates $(x, u, q)$ and $\\theta = du - q\\, dx$.\nThen $d\\theta = dx \\wedge dq$ and $\\theta \\wedge d\\theta = du \\wedge dx \\wedge dq \\ne 0$.\nSo $\\theta$ is a contact form in the sense of [[contact geometry|HBQ8BZ7XB5]], and $J^1\\pi$ is a contact manifold.\n\nNon-example: on the same space the [[form|0E3G0GRPZ6]] $du$ is not a contact form.\nFor $s(x) = x$ one gets $(j^1 s)^* du = dx \\ne 0$.\n\nFor $k = 1$ and one fiber coordinate, $J^1\\pi$ is a contact manifold for every $\\dim M$, with contact form $du - \\sum_i u_i\\, dx^i$.\nFor $k \u003e 1$ or more than one fiber coordinate, the contact distribution has codimension greater than $1$.\nSo it is not a contact structure in the sense of contact geometry, though the name is the same.\n",
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