CR Structures as Integrable G-Structures

Question

Let $M$ be a closed manifold, with dimension $2n+1$. Let $F(M)$ be the frame bundle, a principal $GL(2n+1,\mathbb{R})$-bundle over $M$. An almost CR structure $P$ on $M$ is a structure group reduction of $F(M)$ to the subgroup of $GL(2n+1,\mathbb{R})$ given by $$ \begin{bmatrix} A & x\\ 0 & \tilde x \end{bmatrix}, $$ where $A\in GL(n,\mathbb{C})$, $x\in \mathbb{R}^{2n}$, and $\tilde x\in \mathbb{R}^\times$. (Webster, 1978). Let $G_0$ denote the aforementioned group.

In general, a $G$-structure is integrable if we can select charts on $M$ such that the corresponding frames are elements of the structure group reduction, and integrability of a $G$-structure corresponds to a notion of flatness (Kobayashi). For example, a Riemannian metric is a reduction of the frame bundle to $O(n)$, and an integrable $O(n)$-structure corresponds to a Riemannian metric with vanishing curvature.

Questions: What is the corresponding notion of integrability for almost CR structures? If $M$ has an almost integrable CR structure, is it a hyperplane in $\mathbb{C}^{n+1}$?

Answer 1

I assume that, by 'corresponding frame' for a given chart, you mean the local frame field defined by the coordinate vector fields of the chart. If that is not what you mean, you should specify. Also, in your last sentence, I assume that you meant to write 'integrable almost $CR$-structure' rather than 'almost integrable $CR$-structure'. (I'm not sure what the latter would be anyway.)

Assuming this, you should be aware that your usage of the term 'integrable $G$-structure' is different from the (now standard) usage introduced by S.-s. Chern (Pseudo-groupes continus infinis, Geom. Diffl. Coll. Inter. de C.N.R.S., Strasbourg (1953), 119–136.) For example, in your usage, an integrable $\{e\}$-structure is a framing by vector fields that pairwise Lie-commute, whereas, in Chern's usage, an integrable $\{e\}$-structure is a framing by vector fields such that the Lie-bracket of any two is a constant linear combination of the given vector fields, i.e., the vector fields are the basis of a Lie algebra of vector fields on the manifold. You usage of 'integrable $G$-structure' is what we usually call 'flat $G$-structure'.

If we go with your meaning, then, for your given group $G$, an 'integrable' almost $CR$-structure on $M^{2n+1}$ is locally given by a codimension $1$ foliation of $M$ in which each leaf is endowed with a complex structure (varying smoothly across the leaves, of course). Yes, such a structure is locally equivalent to that induced on a hyperplane in $\mathbb{C}^{n+1}$.