So the equation of a surface with topology S2 can be expanded out in terms of spherical harmonic functions. (I believe). A torus T2 which is just S1xS1 can be expanded out in terms of ordinary harmonics which is just Fourier analysis.

But what about surfaces of higher genus, for example a double torus?

What is the equivalent harmonic functions, or Fourier series for these?

Edit: I am talking about how an arbitrary closed 1D curve can be represented as a Fourier series in terms of sines and cosines (as is done in string theory for example). So in terms of a 2D surface I guess the metric would be defined by its embedding in R3. Wouldn't the series expansion determine the shape of the surface and hence the metric not the other way round?

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    $\begingroup$ Harmonics depend on the metrics. So, define a metric on your chosen genus-2 surface, and we can talk about how a wave propagates on that surface, and the eigenvectors thereof. $\endgroup$ – John Dvorak Aug 1 '16 at 20:43
  • $\begingroup$ I thinks its quite hard to define a metric on a genus-2 surface for example. Maybe it would be done piece-wise? What if the surface was given by an equation like f(x,y,z)=0 for example? $\endgroup$ – zooby Aug 1 '16 at 22:08
  • $\begingroup$ The question of whether the series expansion determines the shape of the surface is known as hearing the shape of a drum. Harmonic analysis on arbitrary surfaces is an interesting piece of mathematics, but there's also another and different generalization of Fourier transforms via Pontryagin duals. In any case, I'm not so sure this is really a physics question. $\endgroup$ – ACuriousMind Aug 2 '16 at 9:35

If your surface has a metric, you can expand in eigenfunctions of the Laplace operator. This is kind of like the spherical harmonic expansion for g = 0 or 1, but it's not as nice. The sphere and torus are both homogeneous spaces, which allows you to use representation theory to organize the expansions. None of the higher genus Riemann surfaces are homogeneous spaces.

  • $\begingroup$ Can you explain a bit more about what you mean by homogeneous spaces? $\endgroup$ – zooby Aug 1 '16 at 22:07
  • $\begingroup$ See the wikipedia article. A homogeneous space is a space with a transitive group action. The space of functions on such spaces is naturally a representation of the group, and its decomposition into irreps is basically the same as spherical harmonic or Fourier expansion. $\endgroup$ – user1504 Aug 2 '16 at 1:06

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