Global analysis, analysis on manifolds Definition

Isospectral manifolds

Builds on Riemannian manifold Spectrum of the Laplace–Beltrami operator Isometry of Riemannian manifolds

Definition

Let $(M_1, g_1)$ and $(M_2, g_2)$ be closed Riemannian manifolds. They are isospectral if they have the same spectrum of the Laplace–Beltrami operator. This means that each real number $\lambda$ has the same multiplicity as an eigenvalue of the Laplacian of $M_1$ and of $M_2$ .

Remarks

An isometry $M_1 \to M_2$ carries eigenfunctions to eigenfunctions, so isometric manifolds are isospectral. By Weyl's law, isospectral manifolds have the same dimension and the same volume. Variants compare the spectra of the Laplacian on $p$ -forms, or the Dirichlet or Neumann spectra of domains with boundary.

Example: Milnor's two flat tori $\R^{16}/\Gamma_1$ and $\R^{16}/\Gamma_2$ , built from two non-isometric even unimodular lattices, are isospectral and not isometric.

Non-example: the round spheres $S^2$ of radius $1$ and of radius $2$ are not isospectral. Their eigenvalues are $k(k+1)$ and $k(k+1)/4$ , so the first nonzero eigenvalues $2$ and $\tfrac12$ differ.

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Isometry of Riemannian manifolds concept
Riemannian manifold concept
Spectrum of the Laplace–Beltrami operator concept
Isospectral manifolds definition

Tag	`W5HF4GMVN9`
URI	`m2://W5HF4GMVN9`
Kind	definition
Writing	W0 stub
MSC	58J53, 58J50
Defines	Isospectral manifolds
Uses	Riemannian manifold, Spectrum of the Laplace–Beltrami operator, Isometry of Riemannian manifolds

Isospectral manifolds

Definition

Remarks

Reading path

Further reading

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