Global analysis, analysis on manifolds Spectral geometry Theorem

Sunada's theorem

Builds on Riemannian manifold Laplace–Beltrami operator Spectrum of the Laplace–Beltrami operator Covering space

Statement

Let $M$ be a compact connected Riemannian manifold, and let $G$ be a finite group that acts freely on $M$ by isometries. Let $H_1$ and $H_2$ be subgroups of $G$ that are almost conjugate: every conjugacy class $C$ of $G$ satisfies $\abs{C \cap H_1} = \abs{C \cap H_2}$ . Give the quotients $M_1 = M/H_1$ and $M_2 = M/H_2$ the Riemannian metrics for which the projections from $M$ are local isometries. Then $M_1$ and $M_2$ are isospectral: their Laplace–Beltrami operators have the same eigenvalues with the same multiplicities.

Remarks

The idea of the proof is to compare the spaces of eigenfunctions. An eigenfunction on $M/H$ with eigenvalue $\lambda$ is the same as an $H$ -invariant eigenfunction on $M$ . So the multiplicity of $\lambda$ on $M/H$ is the dimension of the space of $H$ -fixed vectors in the representation of $G$ on the $\lambda$ -eigenspace of $M$ . Almost conjugate subgroups give permutation representations $\C[G/H_1]$ and $\C[G/H_2]$ with equal characters, and by Frobenius reciprocity these dimensions agree for every representation of $G$ . If $H_1$ and $H_2$ are conjugate in $G$ , then $M_1$ and $M_2$ are isometric, so the theorem is of interest for almost conjugate subgroups that are not conjugate. The projections $M \to M_i$ are covering maps, so $M_1$ and $M_2$ have a common finite cover. This method gave many pairs of isospectral manifolds that are not isometric, including pairs of Riemann surfaces and, by later work, planar domains that are isospectral for the Dirichlet problem.

Going deeper

For a student

The theorem turns a question about the spectrum of the Laplacian into a question in finite group theory. The algebraic input is the same one that produces number fields with the same Dedekind zeta function, which is where almost conjugate subgroups first appeared. Before this method, examples of isospectral non-isometric manifolds were sporadic, such as Milnor's flat tori in dimension $16$ . The theorem shows that the spectrum of the Laplacian does not determine the metric, so spectral geometry asks which properties the spectrum does determine.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Covering space concept
Laplace–Beltrami operator concept
Riemannian manifold concept
Spectrum of the Laplace–Beltrami operator concept
Sunada's theorem statement

Tag	`7ZMWEDNCVQ`
URI	`m2://7ZMWEDNCVQ`
Kind	statement, theorem
Writing	W0 stub
MSC	58J53, 58J50
Status	This copy of the site has no build data. Run `m2 status 7ZMWEDNCVQ`.
About	Isospectral manifolds
Uses	Riemannian manifold, Laplace–Beltrami operator, Spectrum of the Laplace–Beltrami operator, Covering space