# A certain generalization of the Poincare Bendixson theorem

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.

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

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.