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

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

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

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

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

If E\to B 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:

\Omega=d\omega +\omega\wedge \omega,

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

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

For example, the tangent bundle of a Riemannian manifold we have O(n) as the structure group and \Omega^{}_{} 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 \Omega^{}_{} is an alternative description of the curvature tensor, namely in the standard notation for curvature tensor we have

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

Bianchi identities

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

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

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.

See also

See also: Curvature form, Associated bundle, Chern-Simons form, Connection form, Curvature, Curvature of Riemannian manifolds, Curvature tensor, Differential form, Differential geometry, Exterior derivative