I have a given parametrization $X(u,v)$ of a surface $S$ in $\mathbb{R}$. I must prove that it is a Möbius strip. I cannot use graphical means and I am not to reparametrize the surface- essentially, I have only the parametrization and some somewhat qualitative prior knowledge about Möbius strips. It is a differential geometry class and I should use its methods.

I can show that the parameters are $4π$-periodic in one term/coördinate (call it $u$) when the other coördinate $v≠0$ (possibly within some interval, I would have to look), and are $2π$-periodic in $u$ when $v=0$. Moreover, the signum of the normal vector $N$ to the surface varies after a period of $2π$ in $u$. At a given point $N(u_{0}, v_{0}) = -N(u_{0}+2π, v_{0})$, I think (I would have to check, but let us go with it for now).

I am fairly certain that these conditions (if I formulated them correctly) are necessary for $S$ to be a Möbius strip. But are they sufficient? Distilling the question further: what is a set of necessary and sufficient conditions for proving the $S$, given $X$, is a Möbius strip, preferably in line with the aforementioned facts that I have demonstrated? Prove it. How does one prove this sufficiency in particular?

Also, how do I tell the chirality of the Möbius strip $S$ (for it is one, I have on good word) from $X$?

Thank you!


Given your parametrization, you should be able to show that the surface you get really is a rectangle with two of its opposing sides glued together after inversing their orientation. This is the definition of a Möbius band I would try to work with here. Let's say your parametrization $X$ has a rectangle $[a,b] \times [-c,c]$ for domain. Then showing that $X$ has injective differential everywhere and is injective everywhere except on $\{a\} \times [-c,c]$ and $\{b\} \times [-c,c]$ where it satisfies $X(a,v) = X(b,-v)$ for all $v \in [-c,c]$ should be sufficient.


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