 Connection (principal bundle)

This article is about connections on principal bundles. See connection (mathematics) for other types of connections in mathematics.
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 Gconnection on a principal Gbundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.
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.
Contents
Formal Definition
Let π:P→M be a smooth principal Gbundle over a smooth manifold M. Then a principal Gconnection on P is a differential 1form on P with values in the Lie algebra of G which is Gequivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.
In other words, it is an element ω of such that
 where R_{g} denotes right multiplication by g;
 if and X_{ξ} is the vector field on P associated to ξ by differentiating the G action on P, then ω(X_{ξ}) = ξ (identically on P).
Sometimes the term principal Gconnection refers to the pair (P,ω) and ω itself is called the connection form or connection 1form of the principal connection.
Relation to Ehresmann connections
A principal Gconnection 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 from the vertical bundle V of P (with V_{p}=T_{p}(P_{π(p)}) to . It follows that ω determines uniquely a bundle map v:TP→V which is the identity on V. Such a projection v is uniquely determined by its kernel, which is a smooth subbundle H of TP (called the horizontal bundle) such that TP=V⊕H. This is an Ehresmann connection.
Conversely, an Ehresmann connection H⊂TP (or v:TP→V) on P defines a principal Gconnection ω if and only if it is Gequivariant in the sense that H_{pg} = d(R_{g})_{p}(H_{p}).
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 1form on U with values in . If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:M→G is a smooth map, then (sg)^{*}ω = Ad(g)^{1} s^{*}ω+g^{1}dg. The principal connection is uniquely determined by this family of valued 1forms, and these 1forms are also called connection forms or connection 1forms, particularly in older or more physicsoriented literature.
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 valued 1form on P which is not only Gequivariant, 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 1form on M with values in the adjoint bundle
Conversely, any such one form defines (via pullback) a Gequivariant horizontal 1form on P, and the space of principal Gconnections is an affine space for this space of 1forms.
Induced covariant and exterior derivatives
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 over M is isomorphic to the space of Gequivariant Wvalued functions on P. More generally, the space of kforms with values in is identified with the space of Gequivariant and horizontal Wvalued kforms on P. If α is such a kform, then its exterior derivative dα, although Gequivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative d^{ω} from valued kforms on M to valued (k+1)forms on M. In particular, when k=0, we obtain a covariant derivative on .
Curvature form
The curvature form of a principal Gconnection ω is the valued 2form Ω defined by
It is Gequivariant and horizontal, hence corresponds to a 2form on M with values in . The identification of the curvature with this quantity is sometimes called the second structure equation.
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 R^{n}valued 1form on P, should be taken into account. In particular, the torsion form on P, is an R^{n}valued 2form Θ defined by
Θ is Gequivariant and horizontal, and so it descends to a tangentvalued 2form on M, called the torsion. This equation is sometimes called the first structure equation.
References
 Kobayashi, Shoshichi (1957), "Theory of Connections", Ann. Mat. Pura Appl. 43: 119–194, doi:10.1007/BF02411907
 Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), WileyInterscience, ISBN 0471157333
 Kolář, Ivan; Michor, Peter; Slovák, Jan (1993) (PDF), Natural operators in differential geometry, SpringerVerlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf
Categories: Connection (mathematics)
 Fiber bundles
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