Curvature form

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.

(In another convention, 1/2 does not appear.) 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)] 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 :\sigma\Omega(X, Y) = -\omega([X, Y]) = -[X, Y] + h[X, Y] 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 \sigma\in {1, 2} 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:

:,\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.

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

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

Notes

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.

References

1. since $[\backslash omega\; \backslash wedge\; \backslash omega](X,\; Y)\; =\; \backslash frac\{1\}\{2\}([\backslash omega(X),\; \backslash omega(Y)]\; -\; [\backslash omega(Y),\; \backslash omega(X)])$. Here we use also the $\backslash sigma=2$ Kobayashi convention for the exterior derivative of a one form which is then $d\backslash omega(X,\; Y)\; =\; \backslash frac12(X\backslash omega(Y)\; -\; Y\; \backslash omega(X)\; -\; \backslash omega([X,\; Y]))$
2. Proof: $\backslash sigma\backslash Omega(X,\; Y)\; =\; \backslash sigma\; d\backslash omega(X,\; Y)\; =\; X\backslash omega(Y)\; -\; Y\; \backslash omega(X)\; -\; \backslash omega([X,\; Y])\; =\; -\backslash omega([X,\; Y]).$