Manifold hypersurface foliation and Frobenius theorem

[WWGD]

Sorry, I didn't get yet why the above holds true. Thanks.

You're quoting your own post.

 

[cianfa72]

Yes, what about my last question

Why it follows that $f(x,-y) = - f(x,y)$ ?

Is it just because $f(\text{ },)$ actually doesn't depend on its first "slot" (Since the partial derivative w.r.t. it is null) ?

 

[jbergman]

Yes, what about my last question

Is it just because $f(\text{ },)$ actually doesn't depend on its first "slot" (Since the partial derivative w.r.t. it is null) ?

@lavinia is probably more qualified to answer, but this follows from the fact that he claimed that $f(x+1/2, -y) = -f(x, y)$ and the fact that $\partial_x f = 0$.

 

[cianfa72]

Just to help to visualize the mapping of the open Mobius band given by the atlas in post #31, I drew the following picture. $(U_1,Id)$ and $(U_2,f)$ are the two atlas's chart mapping. Vertical red lines's points are identified with a sign flip.