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  "body": "\n## Summary\n\nA CR manifold is a smooth real manifold with a distinguished involutive subbundle of its complexified tangent bundle, see the [[definition|A2VXXBYX6A]].\nThis subbundle plays the role of the holomorphic tangent directions.\nThe model is a real hypersurface in $\\C^n$, such as the sphere $S^{2n-1}$, with the complex tangent directions that lie inside its tangent spaces.\nThe letters CR stand for Cauchy–Riemann.\nA CR structure is what is needed to say which functions satisfy the [[tangential Cauchy–Riemann equations|X7GP9BECD8]].\nCR geometry studies the boundaries of domains in [[several complex variables|BH57D2FBNR]], and its main invariant is the [[Levi form|80MTNXWW80]].\n\n## Explanations\n\n### L1 human\n\nThe boundary of a domain in $\\C^n$ is a real manifold of odd dimension, so it cannot be a complex manifold.\nStill, at each of its points some tangent directions are complex lines: those that are tangent to the boundary together with their rotation by $i$.\nA CR manifold is a manifold that remembers these complex directions.\nFor the unit sphere in $\\C^2$, the complex tangent directions at $(z_1, z_2)$ form the complex line spanned by $(-\\bar z_2, \\bar z_1)$.\n",
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