Talk:Kneser theorem

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

The EoM article " 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. --Ulf Rehmann 18:17, 9 January 2013 (CET)