Ehresmann Connection - Special Cases - Principal Bundles and Principal Connections

Principal Bundles and Principal Connections

Suppose that E is a smooth principal G-bundle over M. Then an Ehresmann connection H on E is said to be a principal (Ehresmann) connection if it is invariant with respect to the G action on E in the sense that

for any e∈E and g∈G; here denotes the differential of the right action of g on E at e.

The one-parameter subgroups of G act vertically on E. The differential of this action allows one to identify the subspace with the Lie algebra g of group G, say by map . The connection form v of the Ehresmann connection may then be viewed as a 1-form ω on E with values in g defined by ω(X)=ι(v(X)).

Thus reinterpreted, the connection form ω satisfies the following two properties:

• It transforms equivariantly under the G action: for all h∈G, where R_h* is the pullback under the right action and Ad is the adjoint representation of G on its Lie algebra.
• It maps vertical vector fields to their associated elements of the Lie algebra: ω(X)=ι(X) for all X∈V.

Conversely, it can be shown that such a g-valued 1-form on a principal bundle generates a horizontal distribution satisfying the aforementioned properties.

Given a local trivialization one can reduce ω to the horizontal vector fields (in this trivialization). It defines a 1-form ω' on B via pullback. The form ω' determines ω completely, but it depends on the choice of trivialization. (This form is often also called a connection form and denoted simply by ω.)