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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $2n + 1$ and let $\\alpha$ be a contact form on $M$.\nSo $\\alpha \\wedge (d\\alpha)^n$ vanishes nowhere and $(M, \\ker \\alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].\nThe *Reeb vector field* of $\\alpha$ is the unique [[vector field|0299VCW1TH]] $R_\\alpha$ on $M$ such that\n$$\\alpha(R_\\alpha) = 1 \\qquad \\text{and} \\qquad \\iota_{R_\\alpha} d\\alpha = 0.$$\n\n## Remarks\n\nExistence and uniqueness: $d\\alpha$ has rank $2n$ at each point, so its kernel is a line, and $\\alpha$ is nonzero on this line because $\\alpha \\wedge (d\\alpha)^n \\ne 0$.\nThe flow of $R_\\alpha$ preserves $\\alpha$, by Cartan's formula $\\mathcal{L}_{R_\\alpha} \\alpha = d(\\alpha(R_\\alpha)) + \\iota_{R_\\alpha} d\\alpha = 0$.\nThe Reeb field depends on $\\alpha$, not only on the contact structure: replacing $\\alpha$ by $h\\alpha$ changes it in general.\n\nExample: on $\\R^3$ with $\\alpha = dz - y\\,dx$, one has $d\\alpha = dx \\wedge dy$, and $R_\\alpha = \\partial_z$, since $\\alpha(\\partial_z) = 1$ and $\\iota_{\\partial_z}(dx \\wedge dy) = 0$.\nOn $S^3 \\subset \\C^2$ with $\\alpha = \\sum_j (x_j\\, dy_j - y_j\\, dx_j)$, the Reeb field is $R_\\alpha = \\sum_j (x_j \\partial_{y_j} - y_j \\partial_{x_j})$, whose flow is $z \\mapsto e^{it} z$.\n\nNon-example: on $\\R^3$ with $\\alpha = dz - y\\,dx$, the field $\\partial_y$ satisfies $\\iota_{\\partial_y} d\\alpha = -dx \\ne 0$ and $\\alpha(\\partial_y) = 0$, so it is not the Reeb field.\n",
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