Reeb foliation

Example

Let $D = \set{u \in \R^2 : \abs{u} < 1}$ be the open unit disc. On $D \times \R$ consider, for each $c \in \R$, the surface $$L_c = \set{(u, t) \in D \times \R : t = c + e^{1/(1 - \abs{u}^2)}}.$$ These surfaces are disjoint graphs over $D$, and they fill $D \times \R$. Together with the boundary cylinder $\partial D \times \R$, they form a codimension one foliation of the closed solid cylinder $\bar D \times \R$. Each $L_c$ is a plane that bends upwards and approaches the boundary cylinder as $\abs{u} \to 1$, since $e^{1/(1 - \abs{u}^2)} \to \infty$. The translation $t \mapsto t + 1$ maps $L_c$ to $L_{c+1}$, so the foliation passes to the quotient solid torus $\bar D \times (\R / \Z)$. This foliated solid torus is a Reeb component. Its boundary torus is one leaf, and every interior leaf is a plane that spirals towards the boundary.

The $3$-sphere is the union of two solid tori glued along their boundary tori. The gluing sends the meridian of each solid torus to the longitude of the other. Putting a Reeb component on each solid torus gives a smooth codimension one foliation of $S^3$, the Reeb foliation. The leaves are one torus and infinitely many planes.