---
tag: ZHX2VYQDZ3
kind: concept
name: Scalar curvature
msc: [53B20, 53C21]
sources:
  - url: https://en.wikipedia.org/wiki/Scalar_curvature
same_as:
  wikidata: Q1147161
  wikipedia: Scalar curvature
---

## Summary

The scalar curvature of a [[Riemannian manifold|7CBT4SM83Y]] is the function obtained by taking the trace of the [[Ricci curvature|4GTNM0WHDH]] with respect to the metric.
It is the simplest curvature invariant: a single number at each point.
It measures how the volume of a small geodesic ball differs from the volume of a Euclidean ball of the same radius.
Questions such as which manifolds admit metrics of positive scalar curvature are a major theme of geometry and topology.

## Explanations

### L1 human

Scalar curvature is the total amount of curving at a point, averaged over all directions.
If it is positive, small balls around the point have slightly less volume than flat balls of the same radius; if negative, slightly more.
For a surface it is twice the Gaussian curvature.
The unit sphere of dimension $n$ has scalar curvature $n(n - 1)$, so the ordinary unit sphere in $\R^3$ has scalar curvature $2$.