Reeblessness and tautness are not equivalent without extra hypotheses

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For a smooth cooriented codimension-one foliation of a closed oriented three-manifold, tautness implies Reeblessness by A Reeb component obstructs tautness, but Reeblessness alone does not imply tautness.

Here is a closed example. On $T^3=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^3$, with coordinates $(x,y,z)$, put $\alpha =\cos (2\pi x)dx+\sin (2\pi x)dy\mathrm{.}$ This is nowhere zero and $\alpha \wedge d\alpha =0$, so its kernel defines a smooth cooriented foliation by The codimension-one Frobenius criterion. The tori $x=0$ and $x=1\mathrm{/}2$ are leaves. On each intervening strip the other leaves satisfy $y+\frac{1}{2\pi }\log \mathrm{\mid }\sin (2\pi x)\mathrm{\mid }=c(\mathrm{m}\mathrm{o}\mathrm{d}1),$ with $z$ free, and are intrinsically cylinders. There are no plane leaves, so no saturated region can have the interior-plane foliation required by Reeb components of a codimension-one foliation. Thus the foliation is Reebless. The compact saturated region $0\le x\le 1\mathrm{/}2$ has the $\alpha$-positive normal pointing inward at both boundary tori. It is a dead-end component in Dead-end components, so the foliation is not taut by A foliation is taut if and only if it has no dead-end component.

Ranz, Corollary 2.13(ii), printed pp.40–41, instead describes a noncompact strip product. Its strips are not compact dead-end components under this page's definition, so that terminology does not justify the closed-manifold comparison. The explicit torus construction above supplies the witness directly. The single-transversal conclusion additionally uses nonempty compact connected ambient manifolds, as stated in A taut foliation of a compact connected manifold has a single closed transversal.