# Connection (principal bundle)

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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*.

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 of

*G*which is

*and*

**G-equivariant***the*

**reproduces***of the*

**Lie algebra generators***on*

**fundamental vector fields***P*.

In other words, it is an element *ω* of such that

- where
*R*_{g}denotes right multiplication by*g*, and is the adjoint representation on (explicitly, ); - 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 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 , where *VP* = ker(d*π*) is the kernel of the tangent mapping which is called the vertical bundle of *P*. 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 *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 .
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}d*g*. The principal connection is uniquely determined by this family of -valued 1-forms, and these 1-forms are also called **connection forms** or **connection 1-forms**, particularly in older or more physics-oriented 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** Template:Harv. 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 1-form on *P* which 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.

## 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 *G*-equivariant *W*-valued functions on *P*. More generally, the space of *k*-forms with values in 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 d*α*, although *G*-equivariant, is no longer horizontal. However, the combination d*α*+*ω*Λ*α* is. This defines an exterior covariant derivative d^{ω} from -valued *k*-forms 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 *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 *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 1-form on *P*, should be taken into account. In particular, the torsion form on *P*, is an **R**^{n}-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*.

## References

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