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Results tagged with foliations
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user 1227
2
votes
Accepted
A 1 dimensional foliation of $\mathbb{R}^4$ with few compact leaves
Yes, there is. Conceptually, imagine a flow on $\mathbb{R}^3$ with a single closed orbit on the unit circle in the plane $z=0$, while every other trajectory has an increasing z-coordinate. It's then e …
- 6,571
12
votes
Accepted
Are limits of compact leaves compact?
Here's a counterexample where the foliation has codimension two. Consider the integral curves of a flow on the 3-torus $\mathbb{R}^3 \, / \,\mathbb{Z}^3$, defined by $x' = 0, y'=\cos 2\pi x, z'=\sin 2 …
- 6,571
11
votes
Foliation with leaves which are and are not dense
Foliations for these surfaces are understood well enough to rule out having both dense and non-dense leaves. … For the definition of a Reeb component and complete details, see this page, especially the "Foliations of Surfaces" section:
http://www.map.mpim-bonn.mpg.de/Foliations …
- 6,571
4
votes
Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbo...
No, because it has a unique leaf that is a torus. That can't be a stable or unstable manifold; one reason is that it can't get either bigger or smaller under the action of the hyperbolic dynamical sys …
- 6,571