# Curvature form

In differential geometry, the **curvature form** describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of the curvature tensor in Riemannian geometry.

## Contents

## Definition

Let *G* be a Lie group with Lie algebra , and *P* → *B* be a principal *G*-bundle. Let ω be an Ehresmann connection on *P* (which is a -valued one-form on *P*).

Then the **curvature form** is the -valued 2-form on *P* defined by

Here stands for exterior derivative, is defined in the article "Lie algebra-valued form" and *D* denotes the exterior covariant derivative. In other terms,

where *X*, *Y* are tangent vectors to *P*.

There is also another expression for Ω:

where *hZ* means the horizontal component of *Z* and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).^{[1]}

A connection is said to be flat if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology. See also: flat vector bundle.

### Curvature form in a vector bundle

If *E* → *B* is a vector bundle, then one can also think of ω as
a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan:

where is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, (so each is a usual 1-form and each is a usual 2-form) then

For example, for the tangent bundle of a Riemannian manifold, the structure group is O(*n*) and Ω is a 2-form with values in the Lie algebra of O(*n*), i.e. the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor, i.e.

using the standard notation for the Riemannian curvature tensor.

## Bianchi identities

If is the canonical vector-valued 1-form on the frame bundle, the torsion of the connection form is the vector-valued 2-form defined by the structure equation

where as above *D* denotes the exterior covariant derivative.

The first Bianchi identity takes the form

The second Bianchi identity takes the form

and is valid more generally for any connection in a principal bundle.

## Notes

- ↑ Proof: We can assume
*X*,*Y*are horizontal (otherwise both side vanish). In that case, this is a consequence of the invariant formula for exterior derivative*d*and the fact ω(Z) is a unique Lie algebra element that generates the vector field*Z*.

## References

- Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.