Existence and uniqueness of maximal connected integral manifolds

By [L1], restrictions of flat charts to coordinate boxes form an open cover of $M$. By [L5], [L6], and [A2], choose a countable such cover $(U_m,{\phi }_m{)}_{m\in \mathbb{N}}$. A plaque through a point of $L_p$ lies in $L_p$: [L9] joins its points within the plaque, and its tangent spaces are $\mathcal{D}$. Hence the plaques of the chosen cover that meet $L_p$ cover $L_p$.

Let $i:N\to M$ be a connected integral manifold with $i(q)=p$, and put $S=i^{-1}(L_p)$. For $r\in N$, choose a flat-chart domain $U$ about $i(r)$ and let $C$ be the connected component of $i^{-1}(U)$ containing $r$. By [L7], $C$ is an open neighborhood of $r$, and [L3] maps it into one plaque. If $r\in S$, that plaque lies in $L_p$; if $r\mathrm{\notin }S$, it lies in a different reachability class. Thus $S$ and its complement are open. Since $N$ is connected and $q\in S$, one has $S=N$, so $i(N)\subseteq L_p$.

Fix one chosen-cover plaque $P_0$ through $p$. If a chosen-cover plaque $P$ lies in $L_p$, then for each $m$ the components of $P\cap U_m$, viewed in the intrinsic Euclidean-slice topology of $P$, form an at most countable family by [L7]. Applying [L3] to the plaque inclusion shows that each such component lies in one plaque of $U_m$, so $P$ meets at most countably many chosen-cover plaques. By [L8], the same is true across all $m$. Starting from $P_0$, take all neighbors at each finite stage and then the union over the countably many finite stages; [L8] makes the resulting family ${\mathcal{P}}_p$ at most countable. Every member lies in $L_p$ by [L9]. Conversely, on each smooth segment of a tangent path the transverse flat-chart coordinates have zero derivative and are constant by [L10], so the path is locally contained in a chosen-cover plaque. Compactness of its parameter interval gives a finite plaque subdivision from $P_0$ to a plaque containing its endpoint. Hence ${\mathcal{P}}_p$ covers $L_p$.

Give each $P\in {\mathcal{P}}_p$ its Euclidean slice coordinates. By [L2], overlapping plaque coordinates have smooth transition maps, so these patches form a smooth atlas. It is countable by step 2.1, and each Euclidean patch has a countable basis, so [L8] makes the induced topology second countable. The inclusion $j_p$ is continuous and injective in these patches; since $M$ is Hausdorff by [L5], distinct points of $L_p$ have disjoint inverse-image neighborhoods, so the induced topology is Hausdorff. It is locally Euclidean by construction. Therefore it is a smooth $k$-manifold, and in plaque coordinates $j_p$ is an injective immersion with tangent image $\mathcal{D}$.

The class $L_p$ is path connected in this topology: a tangent path from $p$ to any of its points admits the finite plaque subdivision used in step 2.1, and is continuous in each plaque chart. Hence $j_p:L_p\hookrightarrow M$ is a connected integral manifold.

The equation $j_p\circ \mathrm{\Phi }=i$ now forces a unique set map $\mathrm{\Phi }:N\to L_p$. For any flat-chart domain $U$, [L3] maps each connected component of $i^{-1}(U)$ into one plaque, and in that plaque chart $\mathrm{\Phi }$ has the same coordinate expression as $i$. Hence $\mathrm{\Phi }$ is smooth, and uniqueness follows from injectivity of $j_p$.

It remains to prove uniqueness of the smooth structure. If $k=0$, every plaque and hence the connected leaf $L_p$ is the singleton $\{p\}$, which has only its unique zero-manifold structure. Assume $k\ge 1$ and let another smooth structure on $L_p$ make $j_p$ a connected integral injective immersion. By [L3], every point has a connected neighborhood in that alternative structure whose image lies in one plaque of step 3.1. The coordinate expression of $j_p$ from that neighborhood to the plaque is between $k$-manifolds and has invertible derivative because both tangent images equal $\mathcal{D}$; [L4] makes it a local diffeomorphism. Thus the alternative charts and the plaque charts are smoothly compatible in both directions, so the structures coincide.

Steps 3.1 and 4.1 give the asserted connected integral leaf structure, steps 1.2 and 4.2 give the universal factorization, and step 4.3 proves uniqueness; consequently $L_p$ is the unique maximal connected integral manifold through $p$.