Utility frequency

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

$\mathrm{\Omega }=d\omega +\frac{1}{2}[\omega ,\omega ]=D\omega .$

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,

$\mathrm{\Omega }(X,Y)=d\omega (X,Y)+[\omega (X),\omega (Y)].$

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:

$\mathrm{\Omega }=d\omega +\omega \wedge \omega ,$

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

${\mathrm{\Omega }}_j^i=d{\omega }_j^i+\sum _k{\omega }_k^i\wedge {\omega }_j^k.$

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.

$R(X,Y)=\mathrm{\Omega }(X,Y),$

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 $\mathrm{\Theta }$ of the connection form $\omega$ is the vector-valued 2-form defined by the structure equation

$\mathrm{\Theta }=d\theta +\omega \wedge \theta =D\theta ,$

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

$D\mathrm{\Theta }=\mathrm{\Omega }\wedge \theta .$

The second Bianchi identity takes the form

$D\mathrm{\Omega }=0$

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

References

• Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.

See also

• Connection (principal bundle)
• Basic introduction to the mathematics of curved spacetime
• Chern-Simons form
• Curvature of Riemannian manifolds
• Gauge theory