Curvature form

Helping orphans the way you would do it

In differential geometry, the curvature form is a generalisation of curvature tensor to an arbitrary principal bundle with connection.

Let be a bundle with structure group the Lie group G and g be the Lie algebra of G.

Assume denotes the 1-form with values in which defines the connection on a bundle. Then the curvature form is the 2-form

here stands for exterior derivative and is the wedge product (it is a bit strange to apply wedge product to forms with values in , but it works the usual way).

For the tangent bundle of a Riemannian manifold we have as the structure group and is the 2-form with values in (which can be thought of as antisymmetric matrices, given an orthonormal basis). In this case the form is an alternative description of the curvature tensor, namely in the stadard notation for curvatur tensor we have .