# Curvature form

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 [itex]E\to B[itex] be a principal G-bundle. Let us denote the Lie algebra of G by [itex]g[itex]. Let [itex]\omega[itex] 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

[itex]\Omega=d\omega +{1\over 2}[\omega,\omega]=D\omega.[itex]

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

[itex]\Omega(X,Y)=d\omega(X,Y) +{1\over 2}[\omega(X),\omega(Y)]. [itex]

If [itex]E\to B[itex] is a fiber bundle with structure group G one can repeat the same for the associated principal G-bundle.

If [itex]E\to B[itex] is a vector bundle then one can also think of [itex]\omega[itex] as about matrix of 1-forms then the above formula takes the following form:

[itex]\Omega=d\omega +\omega\wedge \omega, [itex]

where [itex]\wedge[itex] is the wedge product. More precisely, if [itex]\omega^i_j[itex] and [itex]\Omega^i_j[itex] denote components of [itex]\omega[itex] and [itex]\Omega[itex] correspondingly, (so each [itex]\omega^i_j[itex] is a usual 1-form and each [itex]\Omega^i_j[itex] is a usual 2-form) then

[itex]\Omega^i_j=d\omega^i_j +\sum_k \omega^i_k\wedge\omega^k_j.[itex]

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

[itex]R(X,Y)Z=\Omega^{}_{}(X\wedge Y)Z.[itex]

## Bianchi identities

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

[itex]D\Theta=\Omega\wedge\theta={1\over 2}[\Omega,\theta][itex],

here D denotes the exterior covariant derivative and [itex]\Theta[itex] the torsion.

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

[itex]D\Omega=0.[itex]

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