Smooth foliations tangent to the boundary

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). Let $W$ be a smooth $n$-manifold with boundary, $n\ge 1$, with boundary charts modelled on the half-space ${\mathbb{H}}^n={\mathbb{R}}^{n-1}\times [0,\mathrm{\infty })$ and smooth structure as in Smooth charts, atlases, and structures with boundary and Topological manifolds with boundary; thus $\mathrm{\partial }W$ is a closed embedded smooth $(n-1)$-manifold (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold).

Fix $1\le q\le n$ and split ${\mathbb{H}}^n={\mathbb{R}}^{n-q}\times {\mathbb{H}}^q$. A regular foliation of $W$ tangent to $\mathrm{\partial }W$ of codimension $q$ is a regular foliation atlas for the smooth structure of $W$ (Regular foliation atlases) whose charts are relatively open subsets of ${\mathbb{H}}^n$ and whose transition maps are compatible with the standard decomposition of ${\mathbb{H}}^n$ into the model plaques ${\mathbb{R}}^{n-q}\times \{y\}$, $y\in {\mathbb{H}}^q$; that is, on every overlap the coordinates of the second factor depend only on the second coordinates of the first factor, exactly as for a foliation atlas on a boundaryless manifold, and the plaque decomposition restricts to the half-space.

The model plaques meet $\mathrm{\partial }{\mathbb{H}}^n={\mathbb{R}}^{n-q}\times \mathrm{\partial }{\mathbb{H}}^q$ in the sets ${\mathbb{R}}^{n-q}\times \{y\}$ with $y\in \mathrm{\partial }{\mathbb{H}}^q$. Consequently, near a boundary point of $W$ the leaves of $F$ are the intersections of the model plaques with the half-space; the boundary $\mathrm{\partial }W$ is a union of leaves of $F$, and each such boundary plaque lies in the induced regular foliation of $\mathrm{\partial }W$ of codimension $q-1$; the tangent distribution $TF$ of the foliation (Smooth distributions on a manifold) is tangent to the boundary along $\mathrm{\partial }W$ in the sense that $T_p\mathrm{\partial }W$ contains $D_p=T{F}_p$ for $p\in \mathrm{\partial }W$. Thus the leaf directions have zero normal component there. A leaf of the foliation restricted to the interior $\mathrm{Int}W$ is a leaf of $F$, and it need not meet $\mathrm{\partial }W$.

The pair uses this vocabulary only for $q=1$: the Reeb foliations of the solid torus and its gluing are tangent to the boundary, and the boundary torus of a Reeb component is a single leaf.