A nonsingular Pfaff one-form and its horizontal paths

Let $U\subseteq {\mathbb{R}}^{n+1}$ be open, $n\ge 1$, with coordinates $(x_1,\dots ,x_n,z)$. For $k\in \{1,2\}$, a $C^k$ one-form on $U$ is a field of linear functionals on tangent vectors of the form $\omega ={\sum }_{i=1}^n{a}_i(x,z)d{x}_i+b(x,z)dz$, where every coefficient is $k$ times continuously differentiable. At $q\in U$, it acts by ${\omega }_q(u_1,\dots ,u_n,w)={\sum }_i{a}_i(q)u_i+b(q)w$. It is nonsingular on $U$ when ${\omega }_q\mathrm{\ne }0$ for every $q$; then $H_q:=\ker {\omega }_q$ is an $n$-dimensional subspace of $T_{q}U\cong {\mathbb{R}}^{n+1}$.

For a $C^1$ scalar function $S:W\to \mathbb{R}$ on an open $W\subseteq U$, its differential is the one-form defined by

$d{S}_q(v):=DS(q)v={\sum }_{i=1}^n{\mathrm{\partial }}_{x_i}S(q)u_i+{\mathrm{\partial }}_{z}S(q)w,v=(u_1,\dots ,u_n,w)\in T_{q}W\mathrm{.}$

In particular, if $S\in C^2(W)$, then $dS$ is a $C^1$ one-form.

For one-forms $\alpha ,\beta$, their exterior product is the alternating two-form $(\alpha \wedge \beta )(u,v)=\alpha (u)\beta (v)-\alpha (v)\beta (u)$. For a one-form $\alpha$ and a two-form $\sigma$, $(\alpha \wedge \sigma )(u,v,w)=\alpha (u)\sigma (v,w)-\alpha (v)\sigma (u,w)+\alpha (w)\sigma (u,v)$. For a $C^1$ one-form $\omega ={\sum }_{i=1}^n{a}_{i}d{x}_i+bdz$, its exterior derivative is the continuous two-form

$d\omega ={\sum }_{1\le i<j\le n}({\mathrm{\partial }}_i{a}_j-{\mathrm{\partial }}_j{a}_i)d{x}_i\wedge d{x}_j+{\sum }_{i=1}^n({\mathrm{\partial }}_{i}b-{\mathrm{\partial }}_z{a}_i)d{x}_i\wedge dz\mathrm{.}$

When $\omega$ is $C^2$, the coefficients of $d\omega$ are $C^1$.

For a $C^2$ one-form on this page, the codimension-one Pfaff condition means the pointwise equation $\omega \wedge d\omega =0$.

A piecewise-$C^1$ path is a continuous $\gamma :[s_0,s_1]\to U$ with $s_0\le s_1$ and admitting a finite partition on each closed subinterval of which it is $C^1$. It is $\omega$-horizontal when ${\omega }_{\gamma (t)}(\dot{\gamma }(t))=0$ on the interior of each such subinterval. Endpoints where a derivative is one-sided are included by continuity. For an open $V\subseteq U$, $p$ is horizontally reachable from $q$ within $V$ if a horizontal path in $V$ joins $q$ to $p$. Both signs of time and finite concatenations are allowed by this definition.

On a neighborhood $W$, an integrating factor for $\omega$ is a nowhere-zero $\mu \in C^1(W)$ for which $\mu \omega =dS$ for some $S\in C^2(W)$ with $dS$ nowhere zero. This definition is local; it asserts neither a global potential on $U$ nor a physical identification or calibration of $S$.