I was trying to prove that if $S$ is a non-orientable surface embedded in orientable 3-manifold $M$. Then $S$ does not seperate the normal bundle $NS$ and $M$ as well.
It's easy to check that if $S$ is orientable(for example the cylinder in $\Bbb{R}^3$), $S$ may seperate the normal bundle, I have no idea why the seperating property is related to the orientability of the normal bundle ?