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

\omega

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

=D\omega.

Here

d

*,*

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) .

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

\omega

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

\omega

and

\Omega

corespondently, (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.

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

\Theta

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

D\Omega=0.