Connection (principal bundle)

principal connectionconnectionconnectionsprincipal connections(principal) connectionconnection formconnection on the principal bundleprincipalprincipal H-connection
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.wikipedia
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Ehresmann connection

connectionEhresmannhorizontal lift
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.
Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
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.
Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.

Lie algebra-valued differential form

Lie algebra-valued formLie algebra-valued forms-valued form
Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra \mathfrak g of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.
Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Connection form

connectionconnection one-formvertical subspace
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.
In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object.

Parallel transport

parallelparallel-transporttransported
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.
One generalization is for principal connections.

Connection (vector bundle)

connectionKoszul connectionlinear connection
Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
Then a (principal) connection on F(E) induces a connection on E.

Principal bundle

principalprincipal ''G''-bundleprincipal fiber bundle
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.

Vertical and horizontal bundles

vertical bundlehorizontalhorizontal bundle
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(dπ) is the kernel of the tangent mapping which is called the vertical bundle of P.
Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice then becomes equivalent to the definition of a connection on the principal bundle.

Cartan connection

Cartan geometryconnectionsCartan
Historically, the emergence of the structure equations are found in the development of the Cartan connection.
It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form.

Exterior covariant derivative

covariant exterior derivativeexterior covariant differentiation
This defines an exterior covariant derivative d ω from P\times^G W-valued k-forms on M to P\times^G W-valued (k+1)-forms on M.
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.

Affine connection

connectionaffineaffine connections
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.
Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.

Adjoint bundle

Hence it is basic and so is determined by a 1-form on M with values in the adjoint bundle
A prime example is the curvature of any connection on P which may be regarded as a 2-form on M with values in.

Maurer–Cartan form

Maurer-Cartan formMaurer–Cartan equationMaurer–Cartan 1-form
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.
as a principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique principal connection on the principal bundle

Torsion tensor

torsiontorsion-freetorsion form
In particular, the torsion form on P, is an R n -valued 2-form Θ defined by
This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the generators of the right action in gl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with the adjoint representation on gl(n).

Curvature form

Bianchi identityBianchi identitiescurvature
The curvature form of a principal G-connection ω is the \mathfrak g-valued 2-form Ω defined by

Mathematics

mathematicalmathmathematician
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.

Differentiable manifold

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

Group action (mathematics)

group actionactionorbit
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.

Fiber bundle

structure grouplocal trivializationtrivial bundle
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(dπ) is the kernel of the tangent mapping which is called the vertical bundle of P. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction.

Associated bundle

associated vector bundleassociatedassociated ''X''-bundle
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. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction.

Section (fiber bundle)

sectionsectionsLocal section
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.

Tangent vector

tangent vectorstangent directionstangent
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.

Frame bundle

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

Adjoint representation

adjoint actionadjoint endomorphismAdjoint representation of a Lie group

Fundamental vector field

motivationthe vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P