So I end up with an interesting (at least for me) problem. It is probably very basic and someone out there might have solved this with a pencil behind their ear. Nevertheless, indulge me for a moment while I share my mathematical adventures with all of you and ask for wisdom.

So I have a manifold $R = (\theta, \phi, W(\theta, \phi))$ where $\theta,\phi \in [-\pi,\pi]$ and $W \in \mathbb{R}$. I want to be able to find the metric tensor of this manifold at every point but to do that I need a way to write the function $W(\theta,\phi)$. I was thinking on writing $W$ as a linear combination of orthonormal functions of a complete set $\{f_k\}$ such that:

$$ W(\theta, \phi) = \sum_k c_k f_k (\theta ,\phi) $$

My own mathematical immaturity led me to think about the Spherical Harmonics, however I learned that the angular coordinates used by Spherical Harmonics are not the same as my own. This, as it later became obvious, is because the coordinates in the Spherical Harmonics are (rather unsurprisingly) spherical, whereas my particular system of coordinates seems to be somewhat toroidal.

So the way I see it, I need some kind of "toroidal harmonics". A set of functions defined in the toroidal coordinates that are complete on the "unit Torus" (is that how you call it when only the angles are allowed to vary?) in the same fashion that the spherical harmonics are complete on the unit sphere.

Looking around I found out that there are functions called "Toroidal Functions". That looked promising for a moment but as far as I could tell, it wasn't exactly what I wanted, maybe it was but I didn't manage to understand it.

I got stuck after that and didn't really find any complete set on a Torus. Maybe there aren't any or maybe I don't actually need one and everything I said here is kinda stupid. Anyway, I was hoping to hear you people on this.



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