Yes, you are right. Since the volume form of a plane is non-vanishing, the pull-back form by an immersion is non-vanishing. Then we get a volume form on a Mobius band. It is absurd.
This argument is used to show there is no immersion from an n-dimensional closed manifold to a n-dimensional open manifold ([G] p54).
十月 16th, 2007 at 10:40 上午
把平面的体积形式用这个浸入拉回，成为 Mobius 带上的体积形式，与不可定向性矛盾？
十月 16th, 2007 at 2:58 下午
Yes, you are right. Since the volume form of a plane is non-vanishing, the pull-back form by an immersion is non-vanishing. Then we get a volume form on a Mobius band. It is absurd.
This argument is used to show there is no immersion from an n-dimensional closed manifold to a n-dimensional open manifold ([G] p54).
[G] Mikhael Gromov, Partial Differential Relations. Springer, 1986