View source for Talk:Kneser theorem


Concerning [[Kneser_theorem | this page]], the EoC  did receive the following
comments by email on Jan 8, 2013:

The EoM article "[[Kneser_theorem| Kneser theorem]]" includes this sentence:

:''Kneser also proved that the case when a foliation has no closed leaves is possible only for a non-orientable foliation on a torus and in this case there is a closed line L transversal to the foliation and intersecting all the leaves.''

But this cannot be right, because the well-known irrational-slope
foliation on $T^2 = R^2 / Z^2$ is an orientable foliation with no closed
leaves. (And it is real analytic.)

E.g., such a foliation could have the (constant) tangent vector field given by
$$V(x,y) := (1, \alpha)$$ where $\alpha$ is an irrational number, and $x$ and $y$ are the usual
toral angle coordinates.

Then every leaf is an embedded copy of R that is dense in the torus. 

The statement I quoted above is *also* incorrect where it implies that
a non-orientable foliation *can* have no closed leaves. 

This is impossible: Every non-orientable foliation on the torus has at least 
one closed leaf.

End of email comment. --[[User:Ulf Rehmann|Ulf Rehmann]] 18:17, 9 January 2013 (CET)