Smooth foliated concordance of codimension-one foliations

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be a closed smooth manifold and let $F_0,F_1$ be transversely oriented codimension-one foliations of $M$ with defining forms ${\omega }_0,{\omega }_1$. A smooth foliated concordance from $F_0$ to $F_1$ is a codimension-one regular foliation $G$ of $W:=M\times [0,1]$, transverse to the two boundary slices $M\times \{0\}$ and $M\times \{1\}$, together with a nowhere-vanishing defining $1$-form $\omega$ for $G$ such that for $j=0,1$ the pullback $i_j^{\ast }\omega$ along $i_j(x):=(x,j)$ is a positive smooth multiple of ${\omega }_j$, and hence a defining form for $F_j$ with its prescribed coorientation. By Restriction of a foliation transverse to the boundary each slice $M\times \{j\}$ carries the codimension-one foliation $G{\mathrm{\mid }}_{M\times \{j\}}$ with defining form $i_j^{\ast }\omega$, so the requirement is exactly that these boundary foliations be $F_j$ with the prescribed co- orientations.

The smooth structure on $M\times [0,1]$ is given explicitly by product charts: near $s=0$ use $(x,s)$ and near $s=1$ use $(x,1-s)$, with x a smooth chart of M. These are half-space charts with smooth product transitions, as required by Smooth charts, atlases, and structures with boundary.