# 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.