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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.

1 Definition

Let G be a Lie group and be a principal G bundle. Let us denote by the Lie algebra of G.

Let 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 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 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 and 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 as the structure group and is the 2-form with values in (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

2 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 the torsion.

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

3 See also

• Chern-Simons form
• Curvature of Riemannian manifoldsIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced a way to describe it as a "little m