Connection (principal bundle)

In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points.

It gives rise to (Ehresmann) connections on any fiber bundle associated to

In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold.

Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.

-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on

This means that 1-forms on the total space are canonically isomorphic to

First note that the fundamental vector fields generating the

provide a bundle isomorphism (covering the identity of

is uniquely determined by its kernel, which is a smooth subbundle

The principal connection is uniquely determined by this family of

The quotient space TP/G is also a manifold, and inherits the structure of a fibre bundle over TM which shall be denoted dπ:TP/G→TM.

A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP/G.

Finally, let Γ be a principal connection in this sense.

The horizontal distribution of the connection is the bundle If ω and ω′ are principal connections on a principal bundle P, then the difference ω′ − ω is a

-valued 1-form on P that is not only G-equivariant, but horizontal in the sense that it vanishes on any section of the vertical bundle V of P. Hence it is basic and so is determined by a 1-form on M with values in the adjoint bundle Conversely, any such one form defines (via pullback) a G-equivariant horizontal 1-form on P, and the space of principal G-connections is an affine space for this space of 1-forms.

This is because the adjoint action is trivial since the Lie algebra is Abelian.

For any linear representation W of G there is an associated vector bundle

over M, and a principal connection induces a covariant derivative on any such vector bundle.

This covariant derivative can be defined using the fact that the space of sections of

is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal.

This defines an exterior covariant derivative dω from

The curvature form of a principal G-connection ω is the

-valued 2-form Ω defined by It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in

The identification of the curvature with this quantity is sometimes called the (Cartan's) second structure equation.

[2] Historically, the emergence of the structure equations are found in the development of the Cartan connection.

has a flat connection[1]pg 68 if and only if there exists an open covering

This is useful because it gives a recipe for constructing flat principal

-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.

If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant Rn-valued 1-form on P, should be taken into account.

In particular, the torsion form on P, is an Rn-valued 2-form Θ defined by Θ is G-equivariant and horizontal, and so it descends to a tangent-valued 2-form on M, called the torsion.

A principal bundle connection form may be thought of as a projection operator on the tangent bundle of the principal bundle . The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection .
A connection is equivalently specified by a choice of horizontal subspace for every tangent space to the principal bundle .
A principal bundle connection is required to be compatible with the right group action of on . This can be visualized as the right multiplication taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces interpreted in terms of the connection form leads to its characteristic equivariance properties.