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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of odd dimension $2n + 1$.\nA *contact structure* on $M$ is a smooth field $\\xi$ of hyperplanes $\\xi_x \\subset T_x M$ with the following property.\nEach point has an open neighborhood $U$ and a [[differential $1$-form|0E3G0GRPZ6]] $\\alpha$ on $U$ with $\\xi = \\ker \\alpha$ on $U$ and\n$$\\alpha \\wedge (d\\alpha)^n \\ne 0 \\quad \\text{at every point of } U.$$\nHere $d$ is the [[exterior derivative|0ZHSTAG8ZY]].\nA *contact manifold* is a pair $(M, \\xi)$ with $\\xi$ a contact structure.\nA $1$-form $\\alpha$ defined on all of $M$ with $\\ker \\alpha = \\xi$ is a *contact form* for $\\xi$.\n\n## Remarks\n\nThe condition does not depend on the choice of $\\alpha$: if $\\alpha' = h \\alpha$ with $h$ nowhere zero, then $\\alpha' \\wedge (d\\alpha')^n = h^{n+1} \\alpha \\wedge (d\\alpha)^n$.\nEquivalently, $d\\alpha$ restricted to each hyperplane $\\xi_x$ is nondegenerate.\nA contact form exists globally exactly when $\\xi$ is coorientable.\n\nExample: on $\\R^3$ with coordinates $(x, y, z)$, let $\\alpha = dz - y\\,dx$.\nThen $d\\alpha = dx \\wedge dy$ and $\\alpha \\wedge d\\alpha = dz \\wedge dx \\wedge dy = dx \\wedge dy \\wedge dz \\ne 0$, so $\\ker \\alpha$ is a contact structure, the standard one.\nThe unit sphere $S^{2n+1} \\subset \\C^{n+1}$ with $\\alpha = \\sum_j (x_j\\, dy_j - y_j\\, dx_j)$ restricted to it is another standard example.\n\nNon-example: on $\\R^3$ the form $\\alpha = dz$ has $d\\alpha = 0$, so $\\alpha \\wedge d\\alpha = 0$; its kernel is tangent to the planes $z = \\text{const}$.\nBy the [[Frobenius theorem|5Q4BHBFXJY]], $\\alpha \\wedge d\\alpha = 0$ is exactly the condition for $\\ker \\alpha$ to be tangent to a family of surfaces.\n",
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