In differential geometry, the curvature form describes the curvature of a connection on a principal bundle. It can be considered as an alternative to or a 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: if X, Y are horizontal vector fields on P, then
where hZ means the horizontal component of Z and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).
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 Élie 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
and is valid more generally for any connection in a principal bundle. Two of the four Maxwell's equations, for example, can be viewed as a special case of the first Bianchi identity with zero torsion.
These identities are named after Luigi Bianchi.
- Connection (principal bundle)
- Introduction to the mathematics of general relativity
- Chern–Simons form
- Curvature of Riemannian manifolds
- Gauge theory