Local horizontal inaccessibility and Pfaff integrability

Put $c_i=-a_i\mathrm{/}b$ and $X_i={\mathrm{\partial }}_{x_i}+c_i{\mathrm{\partial }}_z$. These $C^2$ fields form a basis of $\ker \omega$. Their bracket has only a vertical component, $[X_i,X_j{]}^z={\mathrm{\partial }}_i{c}_j+c_i{\mathrm{\partial }}_z{c}_j-{\mathrm{\partial }}_j{c}_i-c_j{\mathrm{\partial }}_z{c}_i$.

Direct coordinate expansion gives $d\omega (X,Y)=X[\omega (Y)]-Y[\omega (X)]-\omega ([X,Y])$ for $[X,Y]=DYX-DXY$. Suppose for contradiction that $[X_i,X_j](q)\mathrm{\ne }0$ at some $q$. For small $t$, let $Y_t=({\varphi }_i^t{)}_{\ast }{X}_j$, so $Y_t(q)=D{\varphi }_i^t({\varphi }_i^{-t}(q))X_j({\varphi }_i^{-t}(q))$. The flow and variational equations give ${\varphi }_i^{-t}(q)=q-t{X}_i(q)+o(t)$ and $D{\varphi }_i^t({\varphi }_i^{-t}(q))=I+tD{X}_i(q)+o(t)$; differentiability of $X_j$ gives $X_j({\varphi }_i^{-t}(q))=X_j(q)-tD{X}_j(q)X_i(q)+o(t)$. Multiplying yields $Y_t(q)=X_j(q)-t[X_i,X_j](q)+o(t)$. Since $[X_i,X_j](q)=v{\mathrm{\partial }}_z$ for some $v\mathrm{\ne }0$ and $b(q)\mathrm{\ne }0$, ${\omega }_q([X_i,X_j](q))=vb(q)\mathrm{\ne }0$; hence ${\omega }_q(Y_t(q))=-tvb(q)+o(t)\mathrm{\ne }0$ for all sufficiently small nonzero $t$, so $Y_t(q)$ is transverse to $\ker {\omega }_q$. Thus $X_1(q),\dots ,X_n(q),Y_t(q)$ are linearly independent.

By continuity, choose an open Euclidean ball $B$ about $q$, contained in the coordinate neighborhood, on which $b$ and the vertical coefficient of the same bracket $[X_i,X_j]$ are both nonzero. For each $r\in B$, repeat step 2.1 at $r$ with a sufficiently small nonzero $t=t_r$. The flow of $Y_t$ is ${\varphi }_i^t\circ {\varphi }_j^s\circ {\varphi }_i^{-t}$, a finite horizontal concatenation with both time signs available. Starting at $r$, compose this flow with those of $X_1,\dots ,X_n$ to obtain a $C^2$ endpoint map ${\mathrm{\Psi }}_r$ near the zero duration vector. Then ${\mathrm{\Psi }}_r(0)=r$ and its derivative has the independent columns $X_1(r),\dots ,X_n(r),Y_t(r)$. Choose $\mathrm{\mid }t\mathrm{\mid }$ and the duration neighborhood small enough that every intermediate segment stays inside $B$, using continuity of local flows and that $r$ is interior to $B$.

At each $r\in B$, apply [L1] to $H_r(s,p)={\mathrm{\Psi }}_r(s)-p$ at $(0,r)$, with ambient dimension $2(n+1)$ and target dimension $n+1$, permuting the input to $(p,s)$ so $s$ is solved. Its solved derivative is invertible by step 3.1. Conditionally, some open neighborhood $O_r\subseteq B$ of $r$ consists of endpoints ${\mathrm{\Psi }}_r(s(p))$, all reachable from $r$ by horizontal paths inside $B$. Reachability inside $B$ is an equivalence relation: constant paths give reflexivity, path reversal gives symmetry, and finite concatenation gives transitivity. Every equivalence class is open in $B$, since each of its points $r$ has $O_r$ in that same class. If there were more than one class, one class and the union of all other classes would be disjoint nonempty open subsets covering $B$. Write $B=B(c,R)$. It is convex: for $x,y\in B$ and $0\le \lambda \le 1$, the triangle inequality gives $\mathrm{\parallel }(1-\lambda )x+\lambda y-c\mathrm{\parallel }\le (1-\lambda )\mathrm{\parallel }x-c\mathrm{\parallel }+\lambda \mathrm{\parallel }y-c\mathrm{\parallel }<R$. Choose a point in each set and join them by their straight segment, which lies in $B$ by this calculation; the preimages of the class and its complement would separate $[0,1]$ into two nonempty relatively open sets, contradicting [F9]. Hence $B$ is one class: every point of $B$ is reachable from $q$ by a path inside $B$, contradicting (a) with $V=B$.

Thus every bracket $[X_i,X_j]$ vanishes by the contradiction at step 4.1. The coordinate identity from step 2.1 gives $d\omega (X_i,X_j)=-\omega ([X_i,X_j])=0$, hence $(\omega \wedge d\omega )({\mathrm{\partial }}_z,X_i,X_j)=0$ for every pair. These evaluations determine the three-form: modulo the $n$-plane $\ker \omega$, the tangent space has dimension one, so every triple reduces by multilinearity and alternation to a multiple of one ${\mathrm{\partial }}_z$ and two horizontal vectors. Therefore $\omega \wedge d\omega =0$. By [F4], condition (b) follows, conditionally on [L1].

Conversely suppose (b) holds at every point of $U$. The exact global reachability lemma [F8] applies to these local first integrals and the declared horizontal paths. For every $q\in U$ and every open neighborhood $V\subseteq U$ of $q$, it gives a point $p\in V$ not horizontally reachable from $q$ even by a path in $U$, and therefore not by a path in $V$. This proves (a), including neighborhoods larger than an individual first-integral chart, and completes the equivalence.