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Concept
Exterior covariant derivative
Summary
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. 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 be the projection to the horizontal subspace.
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
where vi are tangent vectors to P at u.
Suppose that ρ : G → GL(V) is a representation of G on a vector space V. If φ is equivariant in the sense that
where , then Dφ is a tensorial (k + 1)-form on P of the type ρ: it is equivariant and horizontal (a form ψ is horizontal if ψ(v0, ..., vk) = ψ(hv0, ..., hvk).)
By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:
Let be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace. If φ is a tensorial k-form of type ρ, then
where, following the notation in , we wrote
Unlike the usual exterior derivative, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form φ,
where F = ρ(Ω) is the representation in of the curvature two-form Ω. The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism. Note that D2 vanishes for a flat connection (i.e. when Ω = 0).
If ρ : G → GL(Rn), then one can write
where is the matrix with 1 at the (i, j)-th entry and zero on the other entries. The matrix whose entries are 2-forms on P is called the curvature matrix.
Given a smooth real vector bundle E → M with a connection ∇ and rank r, the exterior covariant derivative is a real-linear map on the vector-valued differential forms which are valued in E:
The covariant derivative is such a map for k = 0.
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