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Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a torus, which is homeomorphic to the Cartesian product of two circles: $S^1\times S^1$, the Fourier series should be defined by: $$f(\phi,\theta)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}F(n,m)e^{im\phi}e^{in\phi},$$

where $$F(n,m)=\frac{1}{2\pi}\int_0^{2\pi}\int_0^{2\pi}f(\phi,\theta)e^{-im\phi}e^{-in\phi}~~r(R+r\sin(\phi))d\theta d\phi.$$

Somehow it doesn't seem to be this simple since for the coordinate $\phi$: $e^{-im\phi}$ doesn't seem to be an eigenfunction of the Laplace-Beltrami operator...

Can someone tell me what is wrong with my interpretation?

Thank you

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PML
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  • Surely $e^{-im\phi}$ is an eigenfunction of $\Delta=\frac{\partial^2}{\partial\phi^2}+\frac{\partial^2}{\partial\theta^2}$ on the flat torus up to some normalization? It isn't clear to me that the property of being an eigenfunction is preserved under homeomorphism though, so if you're looking at the LB operator on a non-flat torus... – anon Mar 25 '13 at 21:44
  • @seaturtles Humm, but being that the case, wouldn't the expression of the Fourier Series be different accordingly with which view considered (LP eigenfunctions vs. Representation theory)? Or the view of a torus being homeomorphic to $S^1$×$S^1$ is just valid for a flat torus? (the wiki article doesn't make any distinction...) – PML Mar 25 '13 at 22:05

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