---
tag: 18MQEGN13R
kind: concept
name: Isospectral manifolds
msc: [58J53, 58J50]
sources:
  - url: https://en.wikipedia.org/wiki/Isospectral
---

## 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|QKAFRDPGE4]], the question whether one can [[hear the shape of a drum|X3VH8H1XV3]].
[[Sunada's theorem|7ZMWEDNCVQ]] gives a general method to construct them.

## Explanations

### L1 human

Two isospectral manifolds sound the same: every frequency of one is a frequency of the other, as often.
If two shapes are copies of each other, they certainly sound the same.
The surprise is that some shapes that are not copies of each other also sound the same.
So the list of frequencies does not always determine the shape, although it does determine the dimension, the volume and the total curvature.