---
tag: F4NWZNTRMD
kind: concept
name: Laplace–Beltrami operator
msc: [58J05, 58J50, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator
same_as:
  wikidata: Q1071846
  wikipedia: Laplace–Beltrami operator
---

## Summary

The Laplace–Beltrami operator is the [[Laplacian|0E0B6RA8K0]] of a [[Riemannian manifold|7CBT4SM83Y]]: the divergence of the gradient of a function, both taken with respect to the metric.
It is a second-order elliptic operator that depends only on the metric, not on coordinates.
Its eigenvalues and eigenfunctions form the subject of spectral geometry, and it governs heat flow, waves and harmonic functions on manifolds.

## Explanations

### L1 human

On flat space, the Laplacian of a function compares its value at a point with its average on small spheres around that point.
The Laplace–Beltrami operator does the same thing on a curved space, measuring distances with the curved metric.
On the unit sphere, the function giving the height above the equator is an eigenfunction: its Laplacian is $-2$ times itself.
The vibrations of a curved membrane and the way heat spreads on a curved surface are both described by this operator.