 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 curvature tensor in Riemannian geometry.
Contents
Definition
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,
Curvature form in a vector bundle
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,
Bianchi identities
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.
References
 S.Kobayashi and K.Nomizu, "Foundations of Differential Geometry", Chapters 2 and 3, Vol.I, WileyInterscience.
See also
 Connection (principal bundle)
 Basic introduction to the mathematics of curved spacetime
 ChernSimons form
 Curvature of Riemannian manifolds
 Gauge theory
Various notions of curvature defined in differential geometry Differential geometry of curves Differential geometry of surfaces Riemannian geometry Curvature of connections Categories: Differential geometry
 Curvature (mathematics)
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