A question about topology from an ignorant logician, so please be kind if this is obvious!
We all know that the Klein bottle, unlike the torus, cannot be embedded in 3-space. And we all know (because we were told) that it can be embedded in 4-space. I can see how to embed the torus in 3-space, using only quadratic stuff (I can do that with stuff I learnt at school) but how does one embed the Klein bottle in 4-space? I am asking because I think if I had an equation to pour over I might start to get a feel for how the injection works and what the bottle looks like. In particular, I would like to know if the isometry group of the embedded bottle looks like the isometry group of the embedded torus in 3-space. I'm hoping it does, because that would put flesh on the assertion that there isn't really any self-intersection.
I asked this of a visiting topologist but she didn't know off the top of her head so I'm trusting that this isn't a stupid question!
Best Answer
The parametrization for the Klein bottle provided by Will Brian is \begin{align} x &= (a+b\cos v)\cos u\\ y &= (a+b\cos v)\sin u\\ z &= (b\sin v)\cos(u/2)\\ t &= (b\sin v)\sin(u/2)\\ \end{align} where $a>b>0$.
This leads to the defining conditions: \begin{align} 4a^2(x^2+y^2)&=(a^2-b^2+t^2+x^2+y^2+z^2)^2\\ y(z^2-t^2)&=2txz\\ tyz&>0 \end{align}
The first equation comes from expressing $b \cos v$ both in $x,y$-terms and in $z,t$-terms.
The second equation comes from expressing $\tan u$ both in $x,y$-terms and in $z,t$-terms.
The inequality comes from $(a+b \cos v)(b \sin u \sin v)^2 > 0$.
There may be one other independent inequality also.