# 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

**nabla∇nabla 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.