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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n \\ge 2$.\nLet $M \\subset X$ be a real hypersurface of class $C^2$.\nNear a point $p \\in M$, write $M = \\set{\\rho = 0}$ in local holomorphic coordinates $z = (z_1, \\dots, z_n)$, where $\\rho$ is a real $C^2$ function with $d\\rho \\ne 0$ on $M$.\nThe [[Levi form|80MTNXWW80]] of $M$ at $p$ is the Hermitian form\n$$\\mathcal{L}_p(w) = \\sum_{j,k=1}^{n} \\frac{\\partial^2 \\rho}{\\partial z_j \\partial \\bar z_k}(p)\\, w_j \\bar w_k$$\non the complex tangent space $H_p = \\set{w \\in \\C^n : \\sum_{j} \\frac{\\partial \\rho}{\\partial z_j}(p)\\, w_j = 0}$.\nThe hypersurface $M$ is *Levi-flat* if $\\mathcal{L}_p = 0$ on $H_p$ for every $p \\in M$.\n\n## Remarks\n\nThe condition does not depend on the choice of $\\rho$ or of the coordinates.\nExample: $M = \\set{\\operatorname{Im} z_1 = 0} \\subset \\C^n$ with $\\rho = \\operatorname{Im} z_1 = (z_1 - \\bar z_1)/(2i)$.\nAll mixed derivatives $\\partial^2 \\rho / \\partial z_j \\partial \\bar z_k$ vanish, so the Levi form is $0$ and $M$ is Levi-flat.\nNon-example: the unit sphere $\\set{\\abs{z_1}^2 + \\cdots + \\abs{z_n}^2 = 1}$ with $\\rho = \\abs{z}^2 - 1$.\nIts Levi form is $\\abs{w_1}^2 + \\cdots + \\abs{w_n}^2$, which is not zero on $H_p$, since $H_p \\ne 0$ when $n \\ge 2$.\nFor a smooth hypersurface, an equivalent condition is that the complex tangent distribution $p \\mapsto H_p$ is integrable.\nThen $M$ is locally [[foliated|98HTB7MSPX]] by complex hypersurfaces of $X$; this foliation is called the Levi foliation.\nA Levi-flat hypersurface is a [[CR manifold|ZC78JKZX9S]] of hypersurface type.\n",
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