CR manifold

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In mathematics, a CR manifold is a differentiable manifold M with a preferred complex distribution L, or in other words a subbundle of the complexified tangent bundle

$\mathbb{C}TM = TM\otimes\mathbb{C}.$

It is required to satisfy

$[L,L]\subseteq L$
$L\cap\bar{L}=\{0\}$

The canonical example of a CR manifold is the real $2 n + 1$ sphere as a submanifold of $\mathbb{C}^{n+1}$ . The bundle $L$ described above is given by

$L = \mathbb{C}TS^{2n+1} \cap T^{1,0}\mathbb{C}^{n+1}$

where $T^{1,0}\mathbb{C}^{n+1}$ is the bundle of holomorphic vectors. The real form of this is given by $P=\Re (L\oplus \bar{L})$ , the bundle given at a point $p\in S^{2n+1}$ concretely in terms of the complex structure, $I$ , on $\mathbb{C}^{n+1}$ by