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 $S\_1$.



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> **Questions:**
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> 1. Is there a precise example of this situation of opposite orientation for two nested spheres?
> 2. 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 $\gamma$?
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The motivation for this question is the following [wonderful counterexample of a nongeodesible foliation of torus determined by a non vanishing vector field](https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273689#273689) .


Note that for $n=2$ the [answer to the above question is affirmative.](https://mathoverflow.net/questions/273970/a-cubic-system-with-two-nested-limit-cycles-with-opposite-orientations/274981#274981)


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.



A certain generalization of the Poincare Bendixson theorem
If I'm not mistaken, the [Reeb stability theorem](https://en.wikipedia.org/wiki/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:



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> **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.
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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.


