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
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
If ϕ is a tensorial k-form of type ρ, then where, following the notation in Lie algebra-valued differential form § Operations, 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[clarification needed] in
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).
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 that are valued in E: The covariant derivative is such a map for k = 0.
The exterior covariant derivatives extends this map to general k. There are several equivalent ways to define this object: In the case of the trivial real line bundle ℝ × M → M with its standard connection, vector-valued differential forms and differential forms can be naturally identified with one another, and each of the above definitions coincides with the standard exterior derivative.
Differential forms valued in the vector bundle may be naturally identified with fully anti-symmetric tensorial forms on the total space of the principal bundle.
Depending on how the exterior covariant derivative is formulated, various alternative but equivalent definitions of curvature (some without the language of exterior differentiation) can be obtained.
It is a well-known fact that the composition of the standard exterior derivative with itself is zero: d(dω) = 0.
In the present context, this can be regarded as saying that the standard connection on the trivial line bundle ℝ × M → M has zero curvature.