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 ${\displaystyle {\mathfrak {g}}}$, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a ${\displaystyle {\mathfrak {g}}}$-valued one-form on P).

Then the curvature form is the ${\displaystyle {\mathfrak {g}}}$-valued 2-form on P defined by

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

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

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

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]

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

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 ${\displaystyle \sigma \in \{1,2\}}$ 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:

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

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

${\displaystyle \Omega _{j}^{i}=d{\omega ^{i}}_{j}+\sum _{k}{\omega ^{i}}_{k}\wedge {\omega ^{k}}_{j}.}$

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.

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

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

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

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

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

The second Bianchi identity takes the form

${\displaystyle \,D\Omega =0}$

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

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

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

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

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.

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