Chemistry - 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 $Ω$ 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

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

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

D Ω = 0.

Curvature form

Definition

Bianchi identities

See also