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 curvature tensor in Riemannian geometry.
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Let G be a Lie group with Lie algebra , and P → B be a principal Gbundle. Let ω be an Ehresmann connection on P (which is a valued oneform on P).
Then the curvature form is the valued 2form on P defined by
Here d stands for exterior derivative, is defined by and D denotes the exterior covariant derivative. In other terms,
If E → B is a vector bundle. then one can also think of ω as a matrix of 1forms and the above formula becomes the structure equation:
where is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, (so each is a usual 1form and each is a usual 2form) then
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2form with values in o(n), 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,
If θ is the canonical vectorvalued 1form on the frame bundle, the torsion Θ of the connection form ω is the vectorvalued 2form 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.

