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 ${\mathfrak {g}}$ , 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,^[1]

\,\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^[2]

\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 _{j}^{i}=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, the bulk of general theory of relativity.^{[clarification needed]}

Notes

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

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.

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