CR manifold

Remarks

Example: let $M$ be a smooth real hypersurface of $\C^n$, and let $L_p = T^{1,0}_p \C^n \cap (T_p M \otimes \C)$ for $p \in M$, where $T^{1,0}\C^n$ is spanned by the $\partial / \partial z_j$. This is a subbundle of rank $n - 1$. Clause 1 holds because $T^{1,0} \C^n \cap T^{0,1} \C^n = 0$. Clause 2 holds because $T^{1,0} \C^n$ is involutive and vector fields tangent to $M$ have brackets tangent to $M$. So $M$ is a CR manifold of CR dimension $n - 1$ and CR codimension $1$.

Non-example: on $M = \R$, the complex line bundle $L$ spanned by $\partial / \partial x$ is involutive but satisfies $\overline{L} = L$, so clause 1 fails.

When $\operatorname{rank}_\C L = 1$, clause 2 holds automatically, since $[fX, gX] = (f \, Xg - g \, Xf) X$. When $\dim_\R M = 2 \operatorname{rank}_\C L$, a CR structure is the same as an integrable almost complex structure, by the Newlander–Nirenberg theorem. Some authors call a structure that satisfies only clause 1 an almost CR structure.