E. Cartan's Structure Equation

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 + omega \wedge \omega= D \omega.

(In another convention, 1/2 does not appear.) Here $d$ stands for exterior derivative, cdot \wedge \cdot/math> 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) + omega(X),\omega(Y)/math>
where ''X'', ''Y'' are tangent vectors to ''P''.

There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: $\sigma\Omega(X, Y) = \sigma d\omega(X, Y) = X\omega(Y) - Y \omega(X) - \omega($, Y = -\omega(, Y.
:$\sigma\Omega(X, Y) = -\omega($, Y = -, Y+ h, Y/math>
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 \$ 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 $_j$ and $_j$ denote components of ω and Ω correspondingly, (so each $_j$ is a usual 1-form and each $_j$ is a usual 2-form) then

:$\Omega^i_j = d_j + \sum_k _k \wedge _j.$

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 contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of 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.

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

{{curvature

Differential geometry
Curvature (mathematics)