Exterior covariant derivative

covariant exterior derivativeexterior covariant differentiation
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.wikipedia
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Connection (principal bundle)

principal connectionconnectionconnections
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection. Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces.
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.

Exterior derivative

exterior calculusexterior differentiationdifferentials
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.

Vertical and horizontal bundles

vertical bundlehorizontalhorizontal bundle
Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces.

Vector-valued differential form

tensorial formvector-valued formwith values in
If ϕ is a k-form on P with values in a vector space V, then its exterior covariant derivative Dϕ is a form defined by
:If E is equipped with a connection ∇ then there is a unique covariant exterior derivative

Curvature form

Bianchi identityBianchi identitiescurvature
where F = ρ is the representation in of the curvature two-form Ω.
Here d stands for exterior derivative, is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative.

Mathematics

mathematicalmathmathematician
In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.

Principal bundle

principalprincipal ''G''-bundleprincipal fiber bundle
Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M.

Representation of a Lie group

representationsrepresentation theory of Lie groupsrepresentation
Suppose that ρ : G → GL(V) is a representation of G on a vector space V. When ρ : G → GL(V) is a representation, one can form the associated bundle E = P × ρ V.

Equivariant map

equivariantintertwining operatorintertwiner
If ϕ is equivariant in the sense that

Abuse of notation

Abuse of terminologyabuse notationabuse of language
By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:

Lie algebra-valued differential form

Lie algebra-valued formLie algebra-valued forms-valued form
Let \omega be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace.

Electromagnetic tensor

electromagnetic field tensorfield strength tensorelectromagnetic field strength tensor
The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism.

Electromagnetism

electromagneticelectrodynamicselectromagnetic force
The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism.

Associated bundle

associated vector bundleassociatedassociated ''X''-bundle
When ρ : G → GL(V) is a representation, one can form the associated bundle E = P × ρ V.

Connection form

connectionconnection one-formvertical subspace
Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let \omega be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace. Then the exterior covariant derivative D given by a connection on P induces an exterior covariant derivative (sometimes called the exterior connection) on the associated bundle, this time using the nabla symbol:

Nabla symbol

nablanabla operator
Then the exterior covariant derivative D given by a connection on P induces an exterior covariant derivative (sometimes called the exterior connection) on the associated bundle, this time using the nabla symbol:

Section (fiber bundle)

sectionsectionsLocal section
:Here, Γ denotes the space of local sections of the vector bundle.

Frame bundle

orthonormal frame bundlelinear frame bundletangent frame bundle
Conversely, given a vector bundle E, one can take its frame bundle, which is a principal bundle, and so obtain an exterior covariant differentiation on E (depending on a connection).

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds.

Gauge covariant derivative

covariant derivative(gauge) covariant derivative
A yet more complicated, yet more accurate and geometrically enlightening, approach is to understand that the gauge covariant derivative is (exactly) the same thing as the exterior covariant derivative on a section of an associated bundle for the principal fiber bundle of the gauge theory; and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure.

Connection (vector bundle)

connectionKoszul connectionlinear connection
In fact, given a connection ∇ on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
D denotes the exterior covariant derivative

Torsion tensor

torsiontorsion-freetorsion form
:Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection.