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Covariant exterior derivative preserves tensoriality

For a principal connection, the covariant exterior derivative sends tensorial forms to tensorial forms.

Let π:P→M\pi:P\to M be a with right action Rg:P→PR_g:P\to P, and let ρ:G→GL(V)\rho:G\to \mathrm{GL}(V) be a finite-dimensional representation with differential ρ∗:g→gl(V)\rho_*:\mathfrak g\to \mathfrak{gl}(V) (see ).

A VV-valued kk-form α∈Ωk(P;V)\alpha\in \Omega^k(P;V) is tensorial of type ρ\rho if

  1. (Horizontality) αp(v1,…,vk)=0\alpha_p(v_1,\dots,v_k)=0 whenever at least one viv_i is vertical (i.e. tangent to the fiber), equivalently ιX#α=0\iota_{X^\#}\alpha=0 for all fundamental vertical vector fields X#X^\# (compare and ).
  2. (ρ\rho-equivariance) (Rg)∗α=ρ(g−1)α(R_g)^*\alpha=\rho(g^{-1})\alpha for all g∈Gg\in G (compare ).

Fix a with connection 11-form ω∈Ω1(P;g)\omega\in \Omega^1(P;\mathfrak g) (see ). Let hor:TP→TP\mathrm{hor}:TP\to TP denote the horizontal projection determined by ω\omega. The covariant exterior derivative of a VV-valued kk-form is the operator

Dω:Ωk(P;V)⟶Ωk+1(P;V),(Dωα)(v0,…,vk):=dα(hor v0,…,hor vk).D^\omega:\Omega^k(P;V)\longrightarrow \Omega^{k+1}(P;V), \qquad (D^\omega\alpha)(v_0,\dots,v_k):=d\alpha(\mathrm{hor}\,v_0,\dots,\mathrm{hor}\,v_k).

If α\alpha is tensorial of type ρ\rho, then DωαD^\omega\alpha is also tensorial of type ρ\rho.

In other words, the covariant exterior derivative restricts to a well-defined map

Dω:Ωtensk(P;V)⟶Ωtensk+1(P;V),D^\omega:\Omega^k_{\mathrm{tens}}(P;V)\longrightarrow \Omega^{k+1}_{\mathrm{tens}}(P;V),

where Ωtensk(P;V)\Omega^k_{\mathrm{tens}}(P;V) denotes VV-valued tensorial kk-forms.

Equivalent characterizations

Equivalently, on tensorial forms one may write the global identity on tensorial forms

Dωα=dα+ρ∗(ω)∧α,D^\omega\alpha = d\alpha + \rho_*(\omega)\wedge \alpha,

where the wedge combines gl(V)\mathfrak{gl}(V) acting on VV with the exterior product.

Examples
  1. From equivariant functions to covariant derivatives. A tensorial 00-form of type ρ\rho is just a ρ\rho-equivariant function f:P→Vf:P\to V. Under the identification of such functions with sections of the associated bundle E=P×ρVE=P\times_\rho V (see ), the tensorial 11-form DωfD^\omega f corresponds to the covariant derivative ∇s\nabla s of the associated section s∈Γ(E)s\in \Gamma(E) (see ).
  1. Adjoint-valued forms and the Bianchi identity setting. Taking V=gV=\mathfrak g with the adjoint representation (see ), tensorial g\mathfrak g-valued forms are the same objects that appear in . In particular, the curvature form Fω∈Ω2(P;g)F^\omega\in\Omega^2(P;\mathfrak g) (see ) is tensorial, hence DωFωD^\omega F^\omega is a well-defined tensorial 33-form.
  1. Frame bundle viewpoint. On the frame bundle of a vector bundle (see ), tensorial forms of the defining representation encode tensor fields on the base. The lemma guarantees that applying DωD^\omega to such a tensorial form produces another tensorial form, which is exactly what is needed for defining covariant derivatives of tensor fields via the frame bundle picture (compare ).