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

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

(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,

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

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

$\sigma \mathrm{\Omega }(X,Y)=-\omega ([X,Y])=-[X,Y]+h[X,Y]$

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:

$\mathrm{\Omega }=d\omega +\omega \wedge \omega ,$

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

${\mathrm{\Omega }}_j^i=d{{\omega }^i}_j+\sum _k{{\omega }^i}_k\wedge {{\omega }^k}_j.$

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.

$R(X,Y)=\mathrm{\Omega }(X,Y),$

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

See also: Contracted Bianchi identities

See also: Riemann curvature tensor § Symmetries and identities

If $\theta$ is the canonical vector-valued 1-form on the frame bundle, the torsion $\mathrm{\Theta }$ of the connection form $\omega$ is the vector-valued 2-form defined by the structure equation

$\mathrm{\Theta }=d\theta +\omega \wedge \theta =D\theta ,$

where as above D denotes the exterior covariant derivative. The first Bianchi identity takes the form

$D\mathrm{\Theta }=\mathrm{\Omega }\wedge \theta .$

The second Bianchi identity takes the form

$D\mathrm{\Omega }=0$

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.

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