# Mechanics on a Klein Bottle

JorisL
Hi

Yesterday during a lecture the Klein Bottle manifold was mentioned.
We've been busy with describing the natural way of conducting mechanics on manifolds for a while now.

Does anybody know how an example of a mechanical system which is described by a klein bottle manifold? I've been trying to think of one and googling didn't really result in anything. Just mentions of the klein bottle but no examples.
I have read something about a sliding constraint + another one but I can't find it anymore (my bad).

Are this kind of manifolds used in GR as well?

Kind regards,

Joris

Staff Emeritus
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Are this kind of manifolds used in GR as well?

Space-time manifolds are not chosen a priori in general relativity at the level of the Einstein field equation nor does solving the Einstein field equation fully specify the space-time manifold. However if one is more interested in global properties of general Lorentzian manifolds (such as causal structure), and not in local geometries solving the Einstein field equation, then sure: any smooth 4-manifold with a Lorentzian metric is a valid candidate. For observational/experimental purposes however a Klein bottle would probably not be considered because it is compact and there is a theorem in GR that any compact Lorentzian manifold contains closed time-like curves, not to mention it is non-orientable so a volume 4-form can't be defined on the space-time.

JorisL
Unlike the Mobius strip, which can be constructed using a strip of paper, Klein bottles and tesseracts (a four dimensional cube) are only hypothetical objects, since we have no way of accessing a fourth spatial dimension. The three dimensional models you see of Klein bottles are not the 'real' thing since they self-intersect, which the true object would not do.

http://en.wikipedia.org/wiki/Klein_bottle

http://en.wikipedia.org/wiki/Tesseract

http://en.wikipedia.org/wiki/Möbius_strip

Well, of this I knew. However the lecturer implied that one can think of systems that have a Klein Bottle as their configuration space. I however can't seem to think of one. It's also true that we haven't seen any way to identify the structure of a manifold with a manifold in mechanical systems.

@WannabeNewton
Thanks for this complete and comprehensive answer. I believe I get what you're trying to say. Or will get it when I've put a little more thought into it.

JorisL
*Kick*

I think I've found a system which does this.
A pendulum with a sliding support point, if you get what I mean.
The distance over which to slide is finite (up and down the bar).
I can relate it to the image of the square on this wiki lemma in the part about the construction.

Thanks for the explanations

Edit:
I actually came up with this during a simplified lecture/overview devoted to the nobel prize and the Higgs topic. This kind of system was used as an analog for something. Glad I went because I almost forgot about it.