C¹ codimension-one regular foliations and transverse orientation

Let $M$ be a smooth $n$-manifold, $n\ge 1$ (Smooth manifolds and their smooth charts). A $C^1$ codimension-one foliation atlas on $M$ is a family of charts ${\phi }_{\alpha }=(x_{\alpha },t_{\alpha }):U_{\alpha }\to {\mathbb{R}}^{n-1}\times \mathbb{R},$ with the $U_{\alpha }\subseteq M$ open covering $M$, each ${\phi }_{\alpha }$ a homeomorphism onto its image whose inverse is of class $C^1$ and whose components are of class $C^1$ (Continuously differentiable maps, local inverses, and local diffeomorphisms), such that for every $\alpha ,\beta$ with $U_{\alpha }\cap U_{\beta }\mathrm{\ne }\mathrm{\varnothing }$ the transition map, wherever defined, has the form $(x_{\beta },t_{\beta })=(g_{\beta \alpha }(x_{\alpha },t_{\alpha }),h_{\beta \alpha }(t_{\alpha })),$ where $g_{\beta \alpha }$ is of class $C^1$ and $h_{\beta \alpha }$ is a one-dimensional $C^1$ local diffeomorphism of intervals (Continuously differentiable maps, local inverses, and local diffeomorphisms). Thus the second coordinate of a chart depends only on the old second coordinate, and charts change along the first coordinates arbitrarily inside the slices of constant second coordinate.

Charts of such an atlas are foliation charts. For a foliation chart ${\phi }_{\alpha }$ and $c\in \mathbb{R}$ the set of points of $U_{\alpha }$ whose second coordinate equals $c$ is a slice, and each of its connected components is a plaque of the atlas. Two points of $M$ are said to be plaque-chain equivalent when they can be joined by a finite chain $P_0,\dots ,P_k$ of plaques with $P_{i-1}\cap P_i\mathrm{\ne }\mathrm{\varnothing }$ for $1\le i\le k$; this is an equivalence relation. Its equivalence classes are the leaves of the atlas, and the leaf through a point $p$ is written $L_p$. A $C^1$ codimension-one foliation $F$ of $M$ is the leaf decomposition determined by one such atlas; by C¹ foliation charts preserve plaque equivalence and transverse orientation ↗ the leaf decomposition is unchanged under chart refinement or replacement by a compatible foliation atlas (cross-transitions locally preserve slices). Plaques belong to charts and generally become smaller under refinement. Equip each leaf with the topology generated by relatively open subsets of plaques, and with the $C^1$ plaque coordinates. The certificate below verifies compatibility and Hausdorffness of this intrinsic leaf topology; for a compact leaf a finite plaque-chart cover also gives second countability.

The foliation $F$ is transversely oriented, or co-oriented, when the transverse coordinates can be signed consistently. Concretely, $F$ is transversely oriented when there is a foliation atlas for $M$ presenting $F$ in which every transition germ $h_{\beta \alpha }$ is orientation-preserving: $h_{\beta \alpha }^{\mathrm{\prime }}(t)>0$ at every $t$ in its domain, equivalently each $h_{\beta \alpha }$ is increasing. The signed form of the condition allows one to choose, for every chart of a foliation atlas, a locally constant sign ${\epsilon }_{\alpha }\in \{+,-\}$ and to replace the transverse coordinate $t_{\alpha }$ by ${\epsilon }_{\alpha }{t}_{\alpha }$; all transitions become increasing precisely when, on every nonempty overlap, the two sign choices compensate the sign of $h_{\beta \alpha }^{\mathrm{\prime }}$. The equivalence of these formulations, and the fact that transverse orientability is likewise independent of the atlas, are proved in C¹ foliation charts preserve plaque equivalence and transverse orientation ↗.

In a transversely oriented foliation the two sides of each plaque carry a consistent sign, and the local transversals to the plaques are ordered in a way respected by all plaque transports; this is the only use made of transverse orientation in the C¹ stability block. The remaining regular-foliation items use smooth foliations and the separately stated smooth coorientation definition.