Exterior covariant derivative

Definition

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 T u P = H u ⊕ V u of each tangent space into the horizontal and vertical subspaces. Let h : T u P → H u 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 v_i 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 R g ( u ) = u g , 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, ..., v_k) = ψ(hv₀, ..., hv_k).)

By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:

Let ω be the connection one-form and ρ ( ω ) the representation of the connection in g l ( V ) . That is, ρ ( ω ) is a g l ( V ) -valued form, vanishing on the horizontal subspace. 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 in g l ( V ) 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 e i j is the matrix with 1 at the (i, j)-th entry and zero on the other entries. The matrix F i j 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 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:

Here, Γ denotes the sheaf of local sections of the vector bundle. The extension is made 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 form; thus, it is given on decomposable elements of the space Ω k ( M ; E ) = Γ ( Λ k ( T ∗ M ) ⊗ E ) of E -valued k-forms by

For a section s of E, we also set

where i X is the contraction by X.

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). Identifying tensorial forms and E-valued forms, one may show that

which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds.

Examples