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I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem.

The problem is:

Show that the mapping $F:\mathbb{R}^2\to\mathbb{R}^4$ given by

$$F(x,y)=((r\cos y+a)\cos x,(r\cos y+a)\sin x,r\sin y\cos\frac{x}{2},r\sin y\sin\frac{x}{2}))$$

induces an embedding of the Klein bottle into $\mathbb{R}^4$.

I know that the Klein bottle is defined as a quotient manifold $T^2/G$, where $T^2$ is the 2-torus, $G$ is a group of diffeomorphisms of $T^2$ formed by $\{A,Id\}$, and $A$ is the antipodal map, i.e. $A(p)=-p$.

Besides, I know that $T^2=S^1\times S^1$, where $S^1$ is the unit circle.

Also, I found that $F$ is injective.

However, I don't have a clue about how to attack this problem. I tried googling, but found nothing.

Any help will be appreciated.

YYF
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    $G$ is definitely not injective. Hint: $G$ is doubly periodic with periods $2 \pi$ in both $x$ and $y$ (i.e. $G(x + 2 \pi, y) = G(x, y + 2 \pi) = G(x,y)$.) See if you can get somewhere with that. – bzc Mar 15 '13 at 02:28
  • @BrandonCarter is it periodic with periods $2\pi$? But, there are $\frac{x}{2}$ and $\frac{y}{2}$ in the $\sin$ and $\cos$. Obviously, $\sin(\frac{x+2\pi}{2})=\sin(\frac{x}{2}+\pi)=-\sin(\frac{x}{2})$. So, I am confused... – YYF Mar 15 '13 at 14:31
  • My mistake, it's doubly periodic with periods $4\pi$. – bzc Mar 15 '13 at 15:33
  • i have a cuestion G is a variety?? please help me my friends!!! –  May 30 '13 at 23:51
  • @Y.Fan: I can't see $y/2$ anywhere. So I'd say it's period $4\pi$ in $x$ and $2\pi$ in $y$. – celtschk Jul 06 '13 at 09:09

2 Answers2

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A quick hint:

Remember that $F$ is an immersion if the differential map is injective. Furthermore, $F$ is an embedding if it is an immersion and the mapping $T\mathbb{R}^2$ onto the image of the differential map is a homeomorphism.

Tom
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Your definition of the Klein bottle is wrong (the one you wrote has fixed points).

The thing is, $T^2$ is the orientable double covering of the Klein bottle. The involution you must consider, writing $T=R/Z$, is as follows: $\iota(x,y)=(x+1/2,-y).$

http://math.stanford.edu/~randrade/teaching/old/2012-2013/148/homework/hw3.pdf

If you write the two last components as a complex number, it will help.

Theon Alexander
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