I was reading the Ambrose -- Singer theorem [2] and I came up with a question. The theorem ends with a situation like the following:
Let $M$ be a connected manifold modeled on the banach space $\mathbf V$ and $E \subseteq \mathrm{T}M$ an involutive subbunde, with fibers of type $\mathbf E$ ($\mathbf E \subseteq \mathbf V$ is complemented in $\mathbf V$). By the frobenius theorem, given some point $x \in M$, we can find a maximal integral connected manifold $N$ passing from $x$ [1], that is, a submanifold $N \subseteq M$: the natural inclusion $\jmath : N \longrightarrow M$ is an immersion and the tangent map $\mathrm{T}\jmath$ induces a VB--isomorphism between $\mathrm{T}N$ and $\jmath^*E$.
The theorem proceeds by proving $N = M$ (that is, the maximal integral manifold equals the initial manifold as sets). Then it asserts that the fibers of $E$ and $\mathrm{T}M$ are isomorphic, otherwise the maximal integral manifold would not be all of $M$ [3]
How can we prove the following implication: $$M = N \Longrightarrow \text{distribution $E$ is trivial, with fibers isomorphic to $\mathrm{T}M$}$$ I cannot find any general proof to this, or any counterexample (actually $M$ is a principal bundle $P$ on connected base having the holonomy group $\Phi(u)$ as structure group, $u \in P$).
Even worse the same argument is used in the infinite dimensional case i 've found at [5]
In the case that $M$ is second countable and finite dimensional: Say that $M = N$ and $E$ is not trivial. Then at any $m \in N$, we can find a local chart $(U, \varphi)$ of $M$ over $m$, a split of $\mathbf V = \mathbf E \times \mathbf F$, such that $\varphi: U \longrightarrow U_{\mathbf E} \times U_{\mathbf F} \subseteq \mathbf E \times \mathbf F$ and for any fixed $y \in U_{\mathbf F}$ the map $$ U_{\mathbf E} \ni z \longmapsto \psi_y(z) \;\stackrel{def}= \varphi^{-1}(z,y) \in M$$ is an integral manifold locally. Then the maximal integral manifold (which is endowed with its own differentiable structure) passes from every distinct plaque $\psi_y$, $y \in U_{\mathbf F}$, and hence it is made non-second countable space. It is known that the maximal integral manifold [4] as the leaf of a foliation must be second countable. This is a contradiction.
But what in the case that $M$ is not second countable or infinite dimensional?
My references are a bit standard:
[1] Serge Lang, Differential and Riemannian Manifolds
[2] Kobayashi - Nomizu, Foundations of differential geometry
[3] Shlomo Sternberg, Lectures on differential geometry
[4] Cesar Camacho, Geometric theory of foliations
and the article is the following:
[5] Vassiliou, On the infinite dimensional holonomy theorem
You're reading very sophisticated material, but this point truly is evident. Remember that for every $p\in N$, $T_pN = E_p$ (by definition of integral manifold). so, if $N=M$, this means that $T_pN = T_pM$, and so $E_p=T_pM$. This of course means that $E=TM$.