Exterior covariant derivative
A smooth manifold .
A Lie group with Lie algebra .
A principal -bundle over .
A finite-dimensional representation .
Definition 1. A form is called:
horizontal, if whenever one of the vectors is vertical.
equivariant, if for all , where denotes the right action of on .
We denote by the space of horizontal and equivariant forms. Note that is in general not closed under the ordinary exterior derivative. There is a canonical isomorphism
where is the vector bundle associated to via the representation .
Definition 2. Let be a connection on . The exterior covariant derivative for forms on
is defined by
where denotes the projection to the horizontal subspace defined by .
Some facts are:
Every form in the image of is horizontal. If a form is equivariant, is also equivariant.
The restriction of to can be described in terms of the connection 1-form and the derivative of the representation :
Here we have used the following general notation: if are vector spaces, , and is a linear map, we have .
The previous fact can be used to show
in particular, if is flat.
Definition 3. The exterior covariant derivative for forms on
is the map induced from under the isomorphism .
The connection itself is not in : it is not horizontal.
The curvature of is . Since , we have
The Bianchi identity is .
Under the isomorphism , the curvature corresponds to a 2-form , and the Bianchi-identity corresponds to .
Revised on August 15, 2010 19:59:29
by Toby Bartels