---
tag: 4GTNM0WHDH
kind: concept
name: Ricci curvature
msc: [53B20, 53C21]
sources:
  - url: https://en.wikipedia.org/wiki/Ricci_curvature
same_as:
  wikidata: Q1195879
  wikipedia: Ricci curvature
---

## Summary

The Ricci curvature of a [[Riemannian manifold|7CBT4SM83Y]] is a trace of the [[Riemann curvature tensor|JA29ETCM02]].
It is a symmetric bilinear form on each tangent space.
Let $v$ be a unit vector and $e_2, \ldots, e_n$ an orthonormal basis of $v^\perp$.
Then $\operatorname{Ric}(v, v)$ is the sum of the [[sectional curvatures|1X42JAA349]] of the planes spanned by $v$ and $e_i$.
It controls how the volume of small balls and cones of geodesics differs from the Euclidean value.
It appears in Einstein's equations of general relativity, in the Ricci flow, and in theorems such as [[Myers's theorem|FPMJEX3DSJ]] and the [[Bishop–Gromov inequality|82F5Q4F1S6]].

## Explanations

### L1 human

Ricci curvature is an average of curvatures in all the planes containing a given direction.
Positive Ricci curvature means that geodesics leaving a point in nearby directions tend to come together, so volumes grow more slowly than in flat space.
Negative Ricci curvature means they spread apart and volumes grow faster.
On the unit sphere of dimension $n$, the Ricci curvature is $(n - 1)$ times the metric.