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,[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. 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 _{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.

Notes

since $[\omega \wedge \omega ](X,Y)={\frac {1}{2}}([\omega (X),\omega (Y)]-[\omega (Y),\omega (X)])$ .
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.

Various notions of curvature defined in differential geometry
Differential geometry of curves	Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature
Differential geometry of surfaces	Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form
Riemannian geometry	Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature
Curvature of connections	Curvature form Torsion tensor Cocurvature Holonomy