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  "body": "\n## Definition\n\nLet $(M, L)$ be a [[CR manifold|ZC78JKZX9S]], with $L \\subseteq TM \\otimes \\C$ its CR structure and $\\overline{L}$ the conjugate subbundle.\nA function $f \\colon M \\to \\C$ of class $C^1$ is a *CR function* if $X f = 0$ for every smooth section $X$ of $\\overline{L}$.\n\n## Remarks\n\nThe bundle $L$ plays the role of the holomorphic tangent vectors $\\partial / \\partial z_j$, so $\\overline{L}$ plays the role of the $\\partial / \\partial \\bar z_j$, and the condition says $\\bar\\partial_b f = 0$.\nOne also defines CR distributions by the same equations in the sense of distributions.\n\nExample: let $M \\subset \\C^n$ be a real hypersurface with its induced CR structure, and let $F$ be holomorphic on a neighbourhood of $M$.\nA section $X$ of $\\overline{L}$ is a combination of the $\\partial / \\partial \\bar z_j$ that is tangent to $M$.\nSo $X (F|_M) = (X F)|_M = 0$, and $F|_M$ is a CR function.\n\nNon-example: on the unit sphere $S^3 \\subset \\C^2$, the bundle $\\overline{L}$ is spanned by $\\overline{Z} = z_2 \\, \\partial / \\partial \\bar z_1 - z_1 \\, \\partial / \\partial \\bar z_2$.\nIt is tangent to $S^3$, since $\\overline{Z}(\\abs{z_1}^2 + \\abs{z_2}^2) = z_2 z_1 - z_1 z_2 = 0$.\nFor $f = \\bar z_1$ one finds $\\overline{Z} f = z_2$, which is not identically zero on $S^3$, so $\\bar z_1$ is not a CR function.\n",
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