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

Definition

Let G be a Lie group and be a principal G-bundle. Let us denote the Lie algebra of G by g. Let ω denotes the connection form on E (which is a g-valued one-form on E).

Then the curvature form is the g-valued 2-form on E defined by

Here d stands for exterior derivative, [ * , * ] is the Lie bracket and D denotes the exterior covariant derivative. More precisely,

If is a fiber bundle with structure group G one can repeat the same for the associated principal G-bundle.

If is a vector bundle then one can also think of ω as about matrix of 1-forms then the above formula takes the following form:

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

For example, the tangent bundle of a Riemannian manifold we have O(n) as the structure group and is the 2-form with values in o(n) (which can be thought of as antisymmetric matrices, given an orthonormal basis). In this case the form is an alternative description of the curvature tensor, namely in the standard notation for curvature tensor we have

Bianchi identities

The first Bianchi identity (for a connection with torsion on the frame bundle) takes the form

,

here D denotes the exterior covariant derivative and Θ the torsion.

The second Bianchi identity holds for general bundle with connection and takes the form

DΩ = 0.

See also

• Chern-Simons form
• Curvature of Riemannian manifolds
• Gauge theory