---
tag: JA29ETCM02
kind: concept
name: Riemann curvature tensor
msc: [53B20, 53B05]
specializes: [WX302T93TW]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_curvature_tensor
same_as:
  wikidata: Q855112
  wikipedia: Riemann curvature tensor
---

## Summary

The Riemann curvature tensor of an [[affine connection|JX65KSVHSJ]], usually the Levi-Civita connection of a Riemannian metric, measures how far second covariant derivatives fail to commute.
For the Levi-Civita connection, it vanishes exactly when the manifold is locally isometric to Euclidean space.
Sectional, Ricci and scalar curvature are all obtained from it.
In general relativity it describes the tidal effects of gravity.

## Explanations

### L1 human

On a flat sheet of paper, carrying an arrow around a closed loop without turning it brings the arrow back unchanged.
On a sphere it does not: the arrow comes back rotated.
The Riemann curvature tensor measures this rotation for very small loops, in every pair of directions at every point.
For a round sphere of radius $r$ the effect is the same in every direction and is proportional to $1/r^2$.
For a cylinder it is zero, because a cylinder can be unrolled flat.