Klein Bottle discrete harmonics?

Question

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus.

But how does it work with the Klein Bottle?

That is, once a function $f$ on a Klein Bottle given, how discrete coefficients $f_{lm}$ are calculated from $f$, and reciprocally ($f$ from $f_{lm}$ )?

Answer 1

Trimok's answer is essentially correct and in fact can be strengthened: the eigenfunctions on the (flat) Klein bottle are exactly $$e^{i(2n+1)x}\sin(my),\ e^{i(2n)x}\cos(my).$$

This is a consequence of the following theorem, which is not so hard to prove using the fact that Riemannian covers are local isometries:

If $\hat{M}\to M$ is a regular cover of closed Riemannian manifolds with deck group $\pi$ acting by isometries on $\hat{M}$, the eigenfunctions of $M$ are precisely the $\pi$-invariant eigenfunctions of $\hat{M}$.

If we are interested in any (complete) metric at all on the Klein bottle, then I am not sure what one can say. There may be a rigidity theorem that implies eigenvalues are extremized on the flat Klein bottle.

Answer 2

The 2-sphere and the 2-torus are naturally endowed with an action of some group $G$ ($SO(3)$ in the former case and translations in the latter). Therefore the functions on these manifolds realize a representation of $G$ which can be decomposed into irreducibles. This corresponds to spherical harmonics and Fourier decompositions. It seems to me (though I'm not sure) that there is no natural group action on the Klein bottle, so this appoach wouldn't work.

Alternatively, one can consider these decompositions as being with respect to eigenfunctions of Laplacian, which is itself defined with respect to some riemannian metric (for instance, induced by flat metric of $\mathbb{R}^3$ for the 2-sphere and flat for the 2-torus). So if you have some preferred metric on the Klein bottle (e.g. flat), you can formally write such kind of expansion. However, the eigenbasis of Laplacian ("functions $f_{lm}$") is not known explicitly except probably in the flat case.

Answer 3

I propose a possible answer, while I am not sure at all it is true...

It will correspond, I think, to a flat metric.

For the 2-torus, we have the 2 identifications :

$$x \rightarrow x + 2\pi \space \space and \space \space y \rightarrow y + 2\pi$$

So the torus eigenfunctions are : $$e^{i(nx+my)}$$

Now, the Klein Bottle could be defined from the 2-torus by the involution :

$$ x \rightarrow x + \pi \space \space y \rightarrow - y $$

So the eigenfunctions of the Klein Bottle should be a subset of the eigenfunctions of the torus, compatible with this involution, so they should be :

$$ e^{i(2n+1)x} \sin (my) \space \space , \space \space e^{i(2n)x} \cos (my)$$