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  "body": "\n## Definition\n\nLet $M$ be a smooth real [[submanifold|VKFW27WXEZ]] of $\\C^n$.\nFor $p \\in M$, let $T^{0,1}_p M = (T_p M \\otimes \\C) \\cap T^{0,1}_p \\C^n$, where $T_p M$ is the [[tangent space|WY6K3P4YCG]] and $T^{0,1}_p \\C^n$ is spanned by $\\partial / \\partial \\bar z_1, \\dots, \\partial / \\partial \\bar z_n$.\nA function $f \\colon M \\to \\C$ of class $C^1$ satisfies the *tangential Cauchy–Riemann equations* if $X f = 0$ for every $p \\in M$ and every $X \\in T^{0,1}_p M$.\n\n## Remarks\n\nThese are the [[Cauchy–Riemann equations|9PT772R94A]] restricted to the directions in which a function on $M$ can be differentiated.\nWhen $T^{0,1} M$ has constant rank, it is the conjugate of a CR structure on $M$, and the solutions are the [[CR functions|JFKYYK2CEB]]; one writes the equations as $\\bar\\partial_b f = 0$.\n\nFor a hypersurface $M = \\set{\\rho = 0}$, with $\\rho$ real and $d\\rho \\ne 0$ on $M$, the space $T^{0,1}_p M$ is spanned by the operators\n$$L_{jk} = \\frac{\\partial \\rho}{\\partial \\bar z_j} \\frac{\\partial}{\\partial \\bar z_k} - \\frac{\\partial \\rho}{\\partial \\bar z_k} \\frac{\\partial}{\\partial \\bar z_j}, \\quad 1 \\le j \u003c k \\le n,$$\nso the equations are $L_{jk} f = 0$.\nEach $L_{jk}$ is tangent to $M$ because $L_{jk} \\rho = 0$.\n\nExample: on $S^3 \\subset \\C^2$, with $\\rho = \\abs{z_1}^2 + \\abs{z_2}^2 - 1$, there is one equation $L_{12} f = 0$ with $L_{12} = z_1 \\, \\partial / \\partial \\bar z_2 - z_2 \\, \\partial / \\partial \\bar z_1$.\nThe function $f = z_1 z_2$ is a solution, since it does not depend on $\\bar z_1, \\bar z_2$.\n\nNon-example: $f = \\abs{z_1}^2$ on $S^3$.\nThen $L_{12} f = -z_2 z_1$, which is nonzero at points where $z_1 z_2 \\ne 0$, so $f$ does not satisfy the equations.\n\nFor a real curve $M$ in $\\C$, $T^{0,1} M = 0$, so every function satisfies the tangential equations.\n",
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