The pullback foliation under a transverse map

$D^{\ast }$ is a smooth subbundle. Let $U$ be a foliation chart of $F$ with transverse coordinates $y:U\to {\mathbb{R}}^q$, so $D{\mathrm{\mid }}_U=\ker dy$ by [F2], and put $V:=f^{-1}(U)$, an open subset of $N$. Then $g:=y\circ f:V\to {\mathbb{R}}^q$ is a submersion: for $x\in V$, $d{g}_x=d{y}_{f(x)}\circ d{f}_x$ annihilates $\ker d{f}_x$ and maps $d{f}_x(T_{x}N)$ onto $d{y}_{f(x)}(T_{f(x)}M)=T_{y(f(x))}{\mathbb{R}}^q$, the last surjectivity because $d{f}_x(T_{x}N)+D_{f(x)}=T_{f(x)}M$ and $dy$ kills $D$. Hence $\ker d{g}_x=(d{f}_x{)}^{-1}(\ker d{y}_{f(x)})=D_x^{\ast }$ for every $x\in V$. Thus $D^{\ast }$ is locally the kernel of a constant-rank bundle map $TN{\mathrm{\mid }}_V\to V\times {\mathbb{R}}^q$, and by [F3] it is a smooth subbundle of rank $\dim N-q$ on $V$. The local descriptions agree on overlaps, since all of them compute the same family of subspaces $D_x^{\ast }$; hence $D^{\ast }$ is a smooth distribution of rank $\dim N-q$ on all of $N$.

Local integral manifolds. In step 1.1, $g=y\circ f$ is a submersion. By [F7], locally it is a coordinate projection, so a small connected piece of $g^{-1}(g(x))$ is an embedded submanifold of dimension $\dim N-q$ with tangent space $\ker dg=D^{\ast }$. Thus $D^{\ast }$ has an integral manifold through every point.

Intrinsic leaf preimages. Let $j_L:L\to M$ be the intrinsic integral immersion of a leaf. By [F1], $f$ and $j_L$ are transverse, so [F7] makes $P_L=N{\times }_{M}L$ an embedded manifold in $N\times L$. Projection $P_L\to N$ is injective because $j_L$ is. In a plaque neighborhood of $\mathrm{\ell }\in L$, its local image is a level set of $y\circ f$, so step 2.1 shows that this projection is an immersion with tangent image $D^{\ast }$. This gives the stated intrinsic topology on the set $f^{-1}(L)$. Each connected component of $P_L$ is therefore a connected integral manifold of $D^{\ast }$. Other plaques of $L$ in the same ambient chart are different intrinsic neighborhoods; no single transverse value $c_L$ is assigned to all of $L\cap U$.

Global integrability. By [F4] and step 2.1, $D^{\ast }$ is integrable, and [F2] and Regular foliations and integrable distributions correspond associate to it a regular foliation $f^{\ast }F$ whose leaves are the maximal connected integral manifolds of $D^{\ast }$; by [F5] the leaf through a point is the ${\sim }_{D^{\ast }}$-class of that point.

Every leaf of $f^{\ast }F$ lies in a preimage of a leaf of $F$. Let $L^{\mathrm{\prime }}$ be a leaf of $f^{\ast }F$, with inclusion $i:L^{\mathrm{\prime }}\to N$. For $u\in L^{\mathrm{\prime }}$, $d(f\circ i{)}_u(T_u{L}^{\mathrm{\prime }})=d{f}_u(D_u^{\ast })\subseteq D_{f(u)}$. Since $L^{\mathrm{\prime }}$ is connected, [F6] applied to the smooth map $f\circ i$ shows that $f(L^{\mathrm{\prime }})$ lies in a single leaf $L$ of $F$. Hence $L^{\mathrm{\prime }}\subseteq f^{-1}(L)$.

The leaves are exactly the intrinsic components. Let $C$ be a connected component of $P_L$ and let $x$ be in its image in $N$. By step 3.1 and [F5], this image lies in the pullback leaf $L^{\mathrm{\prime }}$ through $x$. Conversely, step 4.1 and the smooth factorization in [F6] give a smooth map $L^{\mathrm{\prime }}\to L$ lifting $f{\mathrm{\mid }}_{L^{\mathrm{\prime }}}$. Its graph defines a continuous map $L^{\mathrm{\prime }}\to P_L$; its image is connected and meets $C$, hence lies in $C$. Thus $L^{\mathrm{\prime }}$ is exactly the image of $C$. Every point of $P_L$ belongs to such a component, proving the leaf description and the final mapping assertion.