Curvature form

From Academic Kids

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

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

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

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

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

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

<math>\Omega=d\omega +\omega\wedge \omega, <math>

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

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

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

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

Bianchi identities

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

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

here D denotes the exterior covariant derivative and <math>\Theta<math> the torsion.

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

<math>D\Omega=0.<math>

See also

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