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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:

  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 $\gamma$?

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.

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    $\begingroup$ It's 'Bendixson', by the way. $\endgroup$ – LSpice Dec 26 '19 at 16:21
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    $\begingroup$ @LSpice Thank you!I fixed it. $\endgroup$ – Ali Taghavi Dec 26 '19 at 17:32
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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.

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    $\begingroup$ No problem :) I was glad to think about this question. $\endgroup$ – Vadim Alekseev Jan 3 at 17:52
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    $\begingroup$ But belive me I realy forget it. But I realy apprecoate your answer. $\endgroup$ – Ali Taghavi Jan 3 at 17:53
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    $\begingroup$ It's honestly absolutely no problem :) $\endgroup$ – Vadim Alekseev Jan 3 at 17:54
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    $\begingroup$ I will consider a 200 rep for this question to remedy my bad conduct. :) $\endgroup$ – Ali Taghavi Jan 3 at 17:54
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    $\begingroup$ Many thanks, that's extremely nice of you! $\endgroup$ – Vadim Alekseev Jan 4 at 21:17

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