Isospectral manifolds

Summary

Two closed Riemannian manifolds are isospectral if their Laplace–Beltrami operators have the same eigenvalues with the same multiplicities. Isometric manifolds are isospectral, and the converse fails: Milnor found the first counterexample, a pair of flat tori of dimension $16$. Isospectral pairs show the limits of the inverse problem of spectral geometry, the question whether one can hear the shape of a drum. Sunada's theorem gives a general method to construct them.