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, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P.
Then the curvature form is the -valued 2-form on P defined by
Here stands for exterior derivative and is defined in the article "Lie algebra-valued form". In other terms,
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
where hZ means the horizontal component of Z, on the right we identified a vertical vector field and a Lie algebra element generating it, and 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:
where is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, then
For example, for the tangent bundle of a Riemannian manifold, the structure group is O and Ω is a 2-form with values in the Lie algebra of O, 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.

Bianchi identities

If is the canonical vector-valued 1-form on the frame bundle,
the torsion of the connection form
is the vector-valued 2-form defined by the structure equation
where as above D denotes the exterior covariant derivative.
The first Bianchi identity takes the form
The second Bianchi identity takes the form
and is valid more generally for any connection in a principal bundle.