In mathematics, 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. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.
Contents
- Formal definition
- Relation to Ehresmann connections
- Form in a local trivialization
- Bundle of principal connections
- Affine property
- Induced covariant and exterior derivatives
- Curvature form
- Connections on frame bundles and torsion
- References
A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. 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.
Formal definition
Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra
In other words, it is an element ω of
-
Ad g ( R g ∗ ω ) = ω where Rg denotes right multiplication by g, andAd g G (explicitly,Ad g X = d d t g exp ( t X ) g − 1 | t = 0 - if
ξ ∈ g and Xξ is the vector field on P associated to ξ by differentiating the G action on P, then ω(Xξ) = ξ (identically on P).
Sometimes the term principal G-connection refers to the pair (P,ω) and ω itself is called the connection form or connection 1-form of the principal connection.
Relation to Ehresmann connections
A principal G-connection ω on P determines an Ehresmann connection on P in the following way. First note that the fundamental vector fields generating the G action on P provide a bundle isomorphism (covering the identity of P) from the bundle VP to
Conversely, an Ehresmann connection H⊂TP (or v:TP→V) on P defines a principal G-connection ω if and only if it is G-equivariant in the sense that
Form in a local trivialization
A local trivialization of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s*ω of a principal connection is a 1-form on U with values in
Bundle of principal connections
The group G acts on the tangent bundle TP by right translation. 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. Let ρ:TP/G→M be the projection onto M. The fibres of the bundle TP/G under the projection ρ carry an additive structure.
The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). 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. Let q:TP→TP/G be the quotient map. The horizontal distribution of the connection is the bundle
Affine property
If ω and ω' are principal connections on a principal bundle P, then the difference ω' - ω is a
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.
Induced covariant and exterior derivatives
For any linear representation W of G there is an associated vector bundle
Curvature form
The curvature form of a principal G-connection ω is the
It is G-equivariant and horizontal, hence corresponds to a 2-form on M with values in
Connections on frame bundles and torsion
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. This equation is sometimes called the first structure equation.