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In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.

Definition

Let G be a Lie group with Lie algebra \mathfrak g, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a \mathfrak g-valued one-form on P).

Then the curvature form is the \mathfrak g-valued 2-form on P defined by

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

Here d stands for exterior derivative, [\cdot,\cdot] is defined by \( [\alpha \otimes X, \beta \otimes Y] := \alpha \wedge \beta \otimes [X, Y]_\mathfrak{g} \) and D denotes the exterior covariant derivative. In other terms,

\( \,\Omega(X,Y)=d\omega(X,Y) + [\omega(X),\omega(Y)].

Curvature form in a vector bundle

If E → B is a vector bundle. then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation:

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

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, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in o(n), the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor, i.e.

\( \,R(X,Y)=\Omega(X,Y), \)

using the standard notation for the Riemannian curvature tensor,

Bianchi identities

If \theta is the canonical vector-valued 1-form on the frame bundle, the torsion \( \Theta \) of the connection form \( \omega \) is the vector-valued 2-form defined by the structure equation

\Theta=d\theta + \omega\wedge\theta = D\theta,

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

\( D\Theta=\Omega\wedge\theta. \)

The second Bianchi identity takes the form

\( \, D \Omega = 0 \)

and is valid more generally for any connection in a principal bundle.
References

S.Kobayashi and K.Nomizu, "Foundations of Differential Geometry", Chapters 2 and 3, Vol.I, Wiley-Interscience.