A foliations as a G-stucture

Question

According to this Wikipedia entry, a foliation is a particular G-structure whose structure group reduction induced by block matrices. I tried to find more details of this approach to foliations, but could not succeed. Can anybody explain me, how this works? A nice reference would also be helpful.

Answer 1

The formulation you are referring to may be a bit misleading. In fact, a smooth distribution of rank $k$ on a manifold of dimension $n$ can be equivalently described as a G-structure. The relevant subgroup of $GL(n,\mathbb R)$ is the group of block matrices of the form $\begin{pmatrix} A & B \\ 0 & C\end{pmatrix}$ with blocks of sizes $k$ and $n-k$. This is exactly the stabilizer of the $k$-dimensional subspace of $\mathbb R^n$ spanned by the first $k$ unit vectors, and this bascially explains the equivalence of a distribution to a $G$-structure. The principal subbundle of the frame bundle corresponding to $H\subset TM$ consists of linear isomorphisms $\mathbb R^n\to T_xM$ which send the subspace $\mathbb R^k$ to $H_x$.

It is then easy to determine the intrinsic torsion of this $G$-structure. This torsion can be interpreted as a skew symmetric bilinear bundle map $H\times H\to TM/H$ and it is exactly the bundle map induced by the Lie bracket of vector fields. Hence vanishing of the intrinsic torsion of the G-structure is equivalent to involutivity of the distribution $H$. Thus you get an equivalent rephrasing of foliations as integrable G-structures, but I am not sure whether G-structure theory is very helpful for the study of foliations beyond this point.