On the Möbius band a 1 form that kills the tangent spaces to the foliation by circles
pulls back to the Euclidean plane to a form f(x,y)dy that is invariant under the action of the group of covering transformations.
This means that f(x+1/2,-y)=-f(x,y)
If this form is closed then the partial derivative of f with respect to x must be zero so f(x,-y) =-f(x,y).
f(x,y)dy must be identically zero along the equatorial circle and this shows that there is no closed 1 form whose kernel is the tangent spaces to the foliation by circles at all points.
Also since f depends only on y, the differential of its antiderivative is equal to fdy. And since f is an odd function its antiderivative is even and thus descends to a function on the Möbius band.
Comment:This exact analysis extends to the Klein bottle which is made by adding integer translations in the y direction to the group of covering transformations of the Möbius band. The vertical lines of the Möbius band are now made into circles and the Klein bottle is the total space of a circle bundle over the circle.
For those who know some homology theory, the above analysis shows that the vertical circles do not represent non-zero real homology classes since there is no real cohomology class that is dual to them. On the other hand, these circles are not homologous to zero over the integers. They are examples of torsion classes. That is: finite multiples of them (in this case their doubles) are homologous to zero. Real homology has no torsion.
cianfa72 said:
From
Wikipedia there is a complete parametrization of the open Mobius band that is continuous from ##\mathbb R^2## to ##\mathbb R^3##. Then if one restricts it to the domain ##0 \leq u < 2\pi, -1 < v < 1## with the subspace topology, it is 1-1 on the image and continuous as well.
The inverse map is 1-1 on the image and continuous with the subspace topology on it, hence we have established an homeomorphism with the respective subspace topologies.
OK.But the problem was to figure it out on on one's own.