---
tag: ZAJFW1S078
kind: concept
name: Curvature form
msc: [53C05]
sources:
  - url: https://en.wikipedia.org/wiki/Curvature_form
same_as:
  wikidata: Q839832
  wikipedia: Curvature form
---

## Summary

The curvature form of a [[connection on a principal bundle|NJ21PN5ZGJ]] is a $2$-form on the total space.
Its values lie in the Lie algebra of the structure group.
It measures the failure of the horizontal distribution of the connection to be integrable, and it vanishes exactly when the connection is flat.
For connections on vector bundles it becomes a $2$-form with values in endomorphisms, and for the Levi-Civita connection it recovers the Riemann curvature tensor.
Its invariant polynomials give the characteristic forms of Chern–Weil theory.

## Explanations

### L1 human

A connection tells how to move horizontally from fibre to fibre.
If you move around a tiny loop in the base and lift the loop horizontally, you may come back to a different point of the same fibre.
The curvature form records this failure, loop direction by loop direction.
In electromagnetism, where the group is the circle, the curvature form is the electromagnetic field.