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. A principal G-connection on a principal G-bundle over a smooth manifold is a particular type of connection which is compatible with the action of the group .

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 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

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.

Let be a smooth principal G-bundle over a smooth manifold . Then a principal -connection on is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on .

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

  1. where denotes right multiplication by , and is the adjoint representation on (explicitly, );
  2. if and is the vector field on P associated to ξ by differentiating the G action on P, then (identically on ).

Sometimes the term principal -connection refers to the pair and itself is called the connection form or connection 1-form of the principal connection.

Computational remarks

Most known non-trivial computations of principal -connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let , be a principal -bundle over ) This means that 1-forms on the total space are canonically isomorphic to , where is the dual lie algebra, hence -connections are in bijection with .

Relation to Ehresmann connections

A principal -connection on determines an Ehresmann connection on in the following way. First note that the fundamental vector fields generating the action on provide a bundle isomorphism (covering the identity of ) from the bundle to , where is the kernel of the tangent mapping which is called the vertical bundle of . It follows that determines uniquely a bundle map which is the identity on . Such a projection is uniquely determined by its kernel, which is a smooth subbundle of (called the horizontal bundle) such that . This is an Ehresmann connection.

Conversely, an Ehresmann connection (or ) on defines a principal -connection if and only if it is -equivariant in the sense that .

Pull back via trivializing section

A trivializing section of a principal bundle is given by a section s of over an open subset of . Then the pullback s*ω of a principal connection is a 1-form on with values in . If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:MG is a smooth map, then . 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 acts on the tangent bundle 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 :TP/GTM. Let ρ:TP/GM 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/GTM such that Γ : TMTP/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:TPTP/G be the quotient map. The horizontal distribution of the connection is the bundle

We see again the link to the horizontal bundle and thus Ehresmann connection.

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.


Maurer-Cartan connection

For the trivial principal -bundle where , there is a canonical connection[1]pg 49

called the Maurer-Cartan connection. It is defined as follows: for a point define


which is a composition

defining the 1-form. Note that

is the Maurer-Cartan form on the Lie group and .

Trivial bundle

For a trivial principal -bundle , the identity section given by defines a 1-1 correspondence

between connections on and -valued 1-forms on [1]pg 53. For a -valued 1-form on , there is a unique 1-form on such that

  1. for a vertical vector
  2. for any

Then given this 1-form, a connection on can be constructed by taking the sum

giving an actual connection on . This unique 1-form can be constructed by first looking at it restricted to for . Then, is determined by because and we can get by taking

Similarly, the form

defines a 1-form giving the properties 1 and 2 listed above.

Extending this to non-trivial bundles

This statement can be refined[1]pg 55 even further for non-trivial bundles by considering an open covering of with trivializations and transition functions . Then, there is a 1-1 correspondence between connections on and collections of 1-forms

which satisfy

on the intersections for the Maurer-Cartan form on , in matrix form.

Global reformulation of space of connections

For a principal bundle the set of connections in is an affine space[1]pg 57 for the vector space where is the associated adjoint vector bundle. This implies for any two connections there exists a form such that

We denote the set of connections as , or just if the context is clear.

Connection on the complex Hopf-bundle

We[1]pg 94 can construct as a principal -bundle where and is the projection map

Note the Lie algebra of is just the complex plane. The 1-form defined as

forms a connection, which can be checked by verifying the definition. For any fixed we have

and since , we have -invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any we have a short exact sequence

where is defined as

so it acts as scaling in the fiber (which restricts to the corresponding -action). Taking we get

where the second equality follows because we are considering a vertical tangent vector, and . The notation is somewhat confusing, but if we expand out each term

it becomes more clear (where ).

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 dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative dω 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 (Cartan's) second structure equation.[2] Historically, the emergence of the structure equations are found in the development of the Cartan connection. When transposed into the context of Lie groups, the structure equations are known as the Maurer–Cartan equations: they are the same equations, but in a different setting and notation.

Flat connections and characterization of bundles with flat connections

We say that a connection is flat if its curvature form . There is a useful characterization of principal bundles with flat connections; that is, a principal -bundle has a flat connection[1]pg 68 if and only if there exists an open covering with trivializations such that all transition functions

are constant. 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.

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 (Cartan's) first structure equation.

Definition in algebraic geometry

If X is a scheme (or more generally, stack, derived stack, or even prestack), we can associate to it its so-called de Rham stack, denoted XdR. This has the property that a principal G bundle over XdR is the same thing as a G bundle with *flat* connection over X.


  1. ^ a b c d e f Dupont, Johan (August 2003). "Fibre Bundles and Chern-Weil Theory" (PDF). Archived from the original (PDF) on 31 March 2022.
  2. ^ Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J. (1980). "Gravitation, gauge theories and differential geometry". Physics Reports. 66 (6): 213–393. Bibcode:1980PhR....66..213E. doi:10.1016/0370-1573(80)90130-1.