TITLE: A certain generalization of the Poincare Bendixson theorem
QUESTION [4 upvotes]: Assume that we  have  a $n-1$  dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a  foliation $\mathcal{F}$. We  fix an orientation for $D$.(For  $n=2$ we  assume  that it is  orientable, that is a  vector  field generates the  distribution.  For  $n>2$ the  distribution is  automatically  orientable then we  fix  an orientation for $D$. )
The  spheres with radius $1$  and  $2$ are  denoted  by $S_1$ and  $S_2$, respectively.
Assume that these  spheres  determine two leaves  of  the  foliation  and the $D$-orientation on $S_2$ coincide the  standard orientation of  $S_2$ while the  $D$-orientation on $S_1$ opposite the  standard orientation of  $S_1$.

Questions:

Is there a  precise example  of this  situation of  opposite orientation for  two  nested spheres? 
Is it true to say that there  exist a  closed  curve  $\gamma \subseteq \{z\in \mathbb{R}^n \mid 1< \parallel z \parallel <2 \}$ such that the  normall  bundle  of  $\gamma$ is identical to the restriction of $D$ to $\gamma$?


The  motivation for  this  question is  the  following  wonderful counterexample of  a nongeodesible  foliation of  torus determined  by  a  non vanishing vector  field  .
Note  that for  $n=2$ the  answer to the  above  question is  affirmative.
Assume  that $X=P\partial_x +Q\partial_y$ is  a non vanishing  vector  field  on the  punctured plane. Assume  that $C_1,C_2$ are two closed orbits of  $X$ such that $C_1$ lies in the  interior of  $C_2$. Moreover the  flow- orientation of  $C_2$ is anti clockwise and  the  flow  orientation of  $C_1$ is  clockwise. (the  situation in the  above  linked  counter example).  Lets  consider the orthogonal vector  field  $Y=-Q\partial_x+P\partial_y$. Then  the  annular  region $R$ bounded by $C_1,C_2$ is  invariant under the positive  flow  of $Y$. So the Poincare  Bendixson theorem implies that there is a  closed orbit  $\gamma \subseteq R$ for   $Y$. Since  $Y\perp  X$ thus the  normal  bundle  to $\gamma$ is  identical to the  $X$  direction.

REPLY [3 votes]: If I'm not mistaken, the Reeb stability theorem prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. Feuilletages: études géométriques, Théorème 3.1:

Theorem (Reeb global stability). Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres); the local stability also ensures that the orientation is preserved locally.
I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.
{"set_name": "stack_exchange", "score": 4, "question_id": 273963}