# Wave solutions on a quotient manifold

Is there a general rule for, given the solution of the wave equation $\phi$ on a manifold $M$, what the solution will be like for the quotient of the manifold by some group $\Gamma$? In particular what the eigenfunctions $\phi_k(x)$ will be?

The classical example is of course the torus, $\Bbb R^n / \Bbb Z^n$, for which we get the result that $\vec k \propto n/L^n$, rather than being continuous. But is there a general theory for arbitrary manifolds and arbitrary groups?

• Is the group $\Gamma$ assumed to be either discrete or continuous? See also the notion of orbifold. – Qmechanic Apr 17 '17 at 15:25
• I think the group has to be discrete for the quotient manifold to be a manifold itself – Slereah Apr 17 '17 at 15:26