---
tag: QKAFRDPGE4
kind: concept
name: Spectral geometry
msc: [58J50, 58J53, 35P15]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_geometry
same_as:
  wikidata: Q3123529
  wikipedia: Spectral geometry
---

## Summary

Spectral geometry studies how the geometry of a Riemannian manifold relates to the spectra of natural differential operators on it.
The main such operator is the [[Laplace–Beltrami operator|F4NWZNTRMD]].
Direct problems bound or compute eigenvalues from geometric data such as curvature, diameter and volume.
Inverse problems ask how much of the geometry is determined by the [[spectrum|B8TYKV7AA1]], as in the question of [[hearing the shape of a drum|X3VH8H1XV3]].
Its tools include the heat kernel, [[Weyl's law|2JK9YMZV2B]] and trace formulas.

## Explanations

### L1 human

A drum, a bell or a guitar string sounds at certain frequencies fixed by its shape.
Spectral geometry studies this link for curved spaces: the frequencies are the eigenvalues of the Laplacian.
From the frequencies one can hear the dimension, the volume and the total curvature.
But not everything can be heard: there exist [[isospectral manifolds|18MQEGN13R]] that sound the same and have different shapes.