Restriction of a foliation transverse to the boundary

Write $i:\mathrm{\partial }W\hookrightarrow W$ for the inclusion and fix $p\in \mathrm{\partial }W$. By [F4] the subspace $T_p\mathrm{\partial }W$ sits inside $T_{p}W$ as a hyperplane, and the transversality hypothesis reads $T_p\mathrm{\partial }W+T_{p}F=T_{p}W$; since $\omega$ defines $F$, moreover $T_{p}F=\ker {\omega }_p$.

If ${\omega }_p$ vanished on all of $T_p\mathrm{\partial }W$, then $T_p\mathrm{\partial }W\subseteq \ker {\omega }_p=T_{p}F$, so the sum $T_p\mathrm{\partial }W+T_{p}F=T_{p}F$ would have dimension $n$ instead of $n+1$; hence $\omega {\mathrm{\mid }}_{\mathrm{\partial }W}$ is nowhere vanishing, the restricted one-form has $\ker (\omega {\mathrm{\mid }}_{\mathrm{\partial }W}{)}_p=T_p\mathrm{\partial }W\cap T_{p}F$ as its kernel, and the dimension formula for two hyperplanes with sum $T_{p}W$ gives $\dim (T_p\mathrm{\partial }W\cap T_{p}F)=n-1$, which is the dimension count of (i) and the kernel description of (ii).

Pulling back $\omega \wedge d\omega =0$ along $i$ with [F1] and [F2] gives $(\omega {\mathrm{\mid }}_{\mathrm{\partial }W})\wedge d(\omega {\mathrm{\mid }}_{\mathrm{\partial }W})=0$; since $\omega {\mathrm{\mid }}_{\mathrm{\partial }W}$ is nowhere vanishing by step 2.1, [F3] makes its kernel an integrable hyperplane distribution on the smooth $n$-manifold $\mathrm{\partial }W$, and [F5] turns that distribution into a codimension-one regular foliation $F{\mathrm{\mid }}_{\mathrm{\partial }W}$ defined by $\omega {\mathrm{\mid }}_{\mathrm{\partial }W}$, completing (i) and (ii).

A defining form that orients $F$ transversely restricts to the nowhere vanishing form $\omega {\mathrm{\mid }}_{\mathrm{\partial }W}$ of step 2.1, whose kernel is the restricted distribution, so the restricted foliation is transversely oriented by $\omega {\mathrm{\mid }}_{\mathrm{\partial }W}$; this is (iv).

Fix a leaf $L^{\mathrm{\prime }}$ of $F{\mathrm{\mid }}_{\mathrm{\partial }W}$ and $p\in L^{\mathrm{\prime }}$. As a connected manifold tangent to $TF$ and contained in $\mathrm{\partial }W$, the leaf $L^{\mathrm{\prime }}$ lies in a leaf $L$ of $F$ and, being connected, in the intrinsic component $C$ of $L\cap \mathrm{\partial }W$ containing $p$. Conversely, at a point $q$ of $C$ a boundary chart with $\mathrm{\partial }W=\{t=0\}$ and a foliation chart for $F$ present $L$ locally as a level set $\{y=y_0\}$, and because $T_{q}L=T_{q}F$ and $T_q\mathrm{\partial }W$ are transverse the functions $y$ and $t$ have independent differentials at $q$; hence $L\cap \mathrm{\partial }W$ is near $q$ an integral manifold of $\ker (\omega {\mathrm{\mid }}_{\mathrm{\partial }W})$ of dimension $n-1$, that is, a plaque of the restricted foliation, and $C$ is covered by such plaques. The set of points of $C$ lying in the leaf $L^{\mathrm{\prime }}$ is then both open and closed in $C$ and nonempty, so it equals $C$; therefore $L^{\mathrm{\prime }}=C$, which is (iii).

Finally, pulling back the identity $d\omega =\eta \wedge \omega$ along $i$ and applying [F1] and [F2] gives $d(\omega {\mathrm{\mid }}_{\mathrm{\partial }W})=i^{\ast }(d\omega )=(i^{\ast }\eta )\wedge (i^{\ast }\omega )=(\eta {\mathrm{\mid }}_{\mathrm{\partial }W})\wedge (\omega {\mathrm{\mid }}_{\mathrm{\partial }W})$, which is (v); together with steps 2.1, 3.1, 3.2 and 4.1 this proves all five assertions.