---
tag: 41S2EZ0KR3
kind: concept
name: Einstein manifold
msc: [53C25, 53C21, 83C05]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Einstein_manifold
same_as:
  wikidata: Q1309313
  wikipedia: Einstein manifold
---

## Summary

An Einstein manifold is a [[Riemannian manifold|7CBT4SM83Y]] whose [[Ricci curvature|4GTNM0WHDH]] is a constant multiple of the metric.
The condition is the vacuum Einstein equation of general relativity, with a cosmological constant, written for a Riemannian metric.
Einstein metrics are natural candidates for a "best" metric on a manifold, and their existence is a central question of Riemannian geometry.

## Explanations

### L1 human

The Ricci curvature measures how the volume of small balls differs from the flat case, direction by direction.
On an Einstein manifold this effect is the same in every direction and at every point.
The round sphere is the basic example: it looks the same everywhere and in every direction.
Flat space is also Einstein, with constant zero.
A product of two round spheres of different dimensions and the same radius is not Einstein, because the two factors bend volumes by different amounts.