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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $T_\\C M = TM \\otimes_\\R \\C$ be its complexified [[tangent bundle|WY6K3P4YCG]].\nA *CR structure* on $M$ is a complex [[vector subbundle|P4DZM975BV]] $L \\subseteq T_\\C M$ such that:\n\n1. $L \\cap \\overline{L} = 0$, where $\\overline{L}$ is the complex conjugate subbundle;\n2. $L$ is involutive: for all smooth sections $X$, $Y$ of $L$, the [[Lie bracket|68FGQEBD6J]] $[X, Y]$ is a section of $L$.\n\nA *CR manifold* is a smooth manifold with a CR structure.\nThe complex rank of $L$ is its *CR dimension*, and $\\dim_\\R M - 2 \\operatorname{rank}_\\C L$ is its *CR codimension*.\n\n## Remarks\n\nExample: let $M$ be a smooth real hypersurface of $\\C^n$, and let $L_p = T^{1,0}_p \\C^n \\cap (T_p M \\otimes \\C)$ for $p \\in M$, where $T^{1,0}\\C^n$ is spanned by the $\\partial / \\partial z_j$.\nThis is a subbundle of rank $n - 1$.\nClause 1 holds because $T^{1,0} \\C^n \\cap T^{0,1} \\C^n = 0$.\nClause 2 holds because $T^{1,0} \\C^n$ is involutive and vector fields tangent to $M$ have brackets tangent to $M$.\nSo $M$ is a CR manifold of CR dimension $n - 1$ and CR codimension $1$.\n\nNon-example: on $M = \\R$, the complex line bundle $L$ spanned by $\\partial / \\partial x$ is involutive but satisfies $\\overline{L} = L$, so clause 1 fails.\n\nWhen $\\operatorname{rank}_\\C L = 1$, clause 2 holds automatically, since $[fX, gX] = (f \\, Xg - g \\, Xf) X$.\nWhen $\\dim_\\R M = 2 \\operatorname{rank}_\\C L$, a CR structure is the same as an integrable almost complex structure, by the Newlander–Nirenberg theorem.\nSome authors call a structure that satisfies only clause 1 an almost CR structure.\n",
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