In differential geometry, the curvature form describes curvature of principal bundle with connection. It can be considered as an alternative or generalization of curvature tensor.
Let G be a Lie group and be a principal G bundle. Let us denote by g the Lie algebra of G.
Let ω denotes the connection form, a 1-form on E with values in g. Then the curvature form is the 2-form with values in g 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 stadard notation for curvatur tensor we have
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.