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  "body": "\n## Summary\n\nThe tangential Cauchy–Riemann equations on a real submanifold of $\\C^n$ are a restricted form of the [[Cauchy–Riemann equations|9PT772R94A]], see the [[definition|KQWJ8SGJEZ]].\nThey are taken only in the anti-holomorphic directions that are tangent to the submanifold.\nTheir solutions are the [[CR functions|JFKYYK2CEB]], and every restriction of a holomorphic function is one.\nThey form an overdetermined system of first-order linear partial differential equations.\nHans Lewy's study of them led to his example of a linear equation with no solutions.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function satisfies $\\partial f / \\partial \\bar z_j = 0$ in every direction.\nOn a hypersurface, a function is only defined on the hypersurface, so only the derivatives along it make sense.\nThe tangential equations keep those combinations of the $\\partial / \\partial \\bar z_j$ that point along the hypersurface.\nOn a real curve in $\\C$ no such combination is left, so there are no equations at all; on a hypersurface in $\\C^2$ there is one equation.\n",
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