Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

Definition

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

Then the curvature form is the [math]\displaystyle{ \mathfrak g }[/math]-valued 2-form on P defined by

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

(In another convention, 1/2 does not appear.) Here [math]\displaystyle{ d }[/math] stands for exterior derivative, [math]\displaystyle{ [\cdot \wedge \cdot] }[/math] is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

[math]\displaystyle{ \,\Omega(X, Y)= d\omega(X,Y) + {1 \over 2}[\omega(X),\omega(Y)] }[/math]

where X, Y are tangent vectors to P.

There is also another expression for Ω: if X, Y are horizontal vector fields on P, then

[math]\displaystyle{ \sigma\Omega(X, Y) = -\omega([X, Y]) = -[X, Y] + h[X, Y] }[/math]

where hZ means the horizontal component of Z, on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and [math]\displaystyle{ \sigma\in \{1, 2\} }[/math] is the inverse of the normalization factor used by convention in the formula for the exterior derivative.

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.

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:

[math]\displaystyle{ \,\Omega = d\omega + \omega \wedge \omega, }[/math]

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

[math]\displaystyle{ \Omega^i_j = d{\omega^i}_j + \sum_k {\omega^i}_k \wedge {\omega^k}_j. }[/math]

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.

[math]\displaystyle{ \,R(X, Y) = \Omega(X, Y), }[/math]

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

If [math]\displaystyle{ \theta }[/math] is the canonical vector-valued 1-form on the frame bundle, the torsion [math]\displaystyle{ \Theta }[/math] of the connection form [math]\displaystyle{ \omega }[/math] is the vector-valued 2-form defined by the structure equation

[math]\displaystyle{ \Theta = d\theta + \omega\wedge\theta = D\theta, }[/math]

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

[math]\displaystyle{ D\Theta = \Omega\wedge\theta. }[/math]

The second Bianchi identity takes the form

[math]\displaystyle{ \, D \Omega = 0 }[/math]

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

The Bianchi identities can be written in tensor notation as: [math]\displaystyle{ R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m} = 0. }[/math]

The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of general theory of relativity.

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
• Contracted Bianchi identities
• Einstein tensor
• Einstein field equations
• General theory of relativity
• Chern-Simons form
• Curvature of Riemannian manifolds
• Gauge theory