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.
Then the curvature form is the -valued 2-form on P defined by
where X, Y are tangent vectors to P.
There is also another expression for Ω:
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:
where is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, (so each is a usual 1-form and each is a usual 2-form) then
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.
using the standard notation for the Riemannian curvature tensor.
where as above D denotes the exterior covariant derivative.
The first Bianchi identity takes the form
The second Bianchi identity takes the form
- Proof: We can assume X, Y are either vertical or horizontal. Then we can assume X, Y are both horizontal (otherwise both sides vanish since Ω is horizontal). In that case, this is a consequence of the invariant formula for exterior derivative d and the fact ω(Z) is a unique Lie algebra element that generates the vector field Z.
- 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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