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