Dead-end components

Let $F$ be a smooth cooriented codimension-one foliation of a boundaryless manifold $M$. A dead-end component is a compact embedded region $N⊊M$ of ambient dimension with nonempty boundary, saturated by leaves of $F$, whose boundary is a union of leaves and whose positive transverse direction points inward at every boundary point. Equivalently, a positive transverse path starting in $N$ cannot leave $N$. The equivalence follows in a boundary defining coordinate $r\ge 0$: inwardness gives $dr({\gamma }^{\mathrm{\prime }})>0$ at every boundary crossing, so a first exit is impossible; conversely an outward transverse vector gives a short exiting path. Compactness and the boundary charts make the boundary a finite union of compact leaves, as shown in A no-transversal leaf bounds a positive accessibility region with finite inward boundary ↗. A foliation with no such region is dead-end free.

Nonempty boundary excludes an entire connected component of a disconnected ambient manifold, where the inward condition would be vacuous. The definition is componentwise and is unchanged by replacing an existing region by its connected component with boundary. The open formulation in other treatments describes the interior of a trapping region; it is not used to assert a compact closure without proof.