Transversely oriented codimension-one foliations

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair, Smooth partitions of unity exist on manifolds). Let $F$ be a regular codimension-one foliation of a smooth manifold $M$ (Regular foliation atlases) with tangent distribution $D=TF$, a smooth rank-$(n-1)$ subbundle of $TM$ (Smooth distributions on a manifold).

The foliation $F$ is transversely oriented, or co-oriented, when there is a nowhere-vanishing smooth $1$-form $\omega$ on $M$ whose kernel is $D$: $D=\ker \omega \mathrm{.}$ Equivalently, $F$ is transversely oriented when there is a nowhere-vanishing smooth vector field $X$ on $M$ transverse to $F$, that is, $X_p\mathrm{\notin }{D}_p$ for every $p\in M$.

The two formulations are equivalent because either object is a trivialization of the same line bundle. A smooth $1$-form vanishing on $D$ is a section of the annihilator bundle $D^{\circ }\subseteq T^{\ast }M$ (The annihilator bundle of a distribution), which has rank one; vanishing of this section is an intrinsic condition, so a nowhere-vanishing $\omega$ with $D=\ker \omega$ is exactly a global frame of $D^{\circ }$, and a line bundle admits a nowhere-vanishing section exactly when it is trivial. Dually, a vector field $X$ transverse to $F$ descends to a nowhere-vanishing section of the normal line bundle $TM\mathrm{/}D$, with the normalizations $\omega (X)=1$ identifying the two trivializations pointwise. Thus transverse orientability is exactly triviality of the normal line bundle $TM\mathrm{/}D$. When $M$ is closed this triviality is a genuine restriction, related to orientability of $M$ and of the foliation (Orientable manifolds).

On a transversely oriented codimension-one foliation the local transversals to $F$ are ordered: in a foliation chart the sign of $\omega$ orients the one-dimensional transverse coordinate, and this orientation is respected by all plaque transports, so the transverse direction is globally coherent along each leaf. This definition concerns smooth foliations; the C¹ block uses C¹ codimension-one regular foliations and transverse orientation instead.

The passage between the two smooth global objects uses only the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ partition-of-unity input (Smooth partitions of unity exist on manifolds). Given $\omega$, locally choose smooth transverse fields $X_i$ with $\omega (X_i)=1$ and patch them with a subordinate partition ${\rho }_i$; then $X={\sum }_i{\rho }_i{X}_i$ satisfies $\omega (X)=1$. Conversely, given transverse $X$, choose local annihilator forms ${\omega }_i$ normalized by ${\omega }_i(X)=1$ and patch them the same way. Their sum annihilates $D$ and evaluates to one on $X$, so its kernel is exactly $D$. This supplies the lift from the normal line to actual smooth fields/forms rather than treating the lift as automatic.