Curvature form

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 the 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

$\Omega=d\omega +{1\over 2}[\omega \wedge \omega]=D\omega.$

Here $d$ stands for exterior derivative, $[\cdot \wedge \cdot]$ is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

$\,\Omega(X,Y)=d\omega(X,Y) + {1 \over 2}([\omega(X),\omega(Y)] - [\omega(Y),\omega(X)])= d\omega(X,Y) + [\omega(X),\omega(Y)]$

where X, Y are tangent vectors to P.

There is also another expression for Ω:

$2\Omega(X, Y) = -[hX, hY] + h[X, Y]$

where hZ means the horizontal component of Z and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).

A connection is said to be flat if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology. See also: flat vector bundle.

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 of E. Cartan:

$\,\Omega=d\omega +\omega\wedge \omega,$

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 the Lie algebra of O(n), i.e. 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

• 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