Exterior covariant derivative
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 so that it gives a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection.
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 V is a representation of G; i.e., there is a Lie group homomorphism ρ: G →GL(V). If φ is equivariant in the sense:
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).)
- Example: if ω is the connection form on P, then Ω = Dω is called the curvature form of ω. Bianchi's second identity says the exterior covariant derivative of Ω is zero; i.e., DΩ = 0.
We also denote the differential of ρ at the identity element by ρ:
If φ is a tensorial k-form of type ρ, then
where is a -valued form, and
- Example: Bianchi's second identity (DΩ = 0) can be stated as: .
Unlike the usual exterior derivative, which squares to 0 (that is d2 = 0), we have:
where F = ρ(Ω). In particular D2 vanishes for a flat connection (i.e., 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.
Exterior covariant derivative for vector bundles
When ρ: G →GL(V) is a representation, one can form the associated bundle E = P ⊗ρ V. Then the exterior covariant differentiation D given by a connection on P defines
through the correspondence between E-valued forms and tensorial forms of type ρ (see tensorial forms on principal bundles.) Requiring ∇ to satisfy Leibniz's rule, ∇ also acts on any E-valued forms. This ∇ is called the exterior covariant differentiation on E. One also sets: for a section s of E,
where is the contraction by X. Explicitly,
since when .
Conversely, given a vector bundle E, one can take its frame bundle, which is a principal bundle, and so gets an exterior covariant differentiation on E (depending on a connection). Identifying tensorial forms and E-valued forms, there is, for example,
- If k = 0, then, writing for the fundamental vector field (i.e., vertical vector field) generated by X in on P, we have:
- Proof: We have:
- Kobayashi, Shoshichi and Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience. ISBN 0-471-15733-3.
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