exterior covariant derivative



Definition on vector bundles

Let p:EMp: E \to M be a vector bundle with a linear connection given by the covariant derivative :Γ(TM)×Γ(E)Γ(E)\nabla: \Gamma(T M) \times \Gamma(E) \to \Gamma(E). We let Ω(M,E)\Omega(M, E) be the space of all differential forms with values in EE. To define the exterior covariant derivative, we take the explicit formula of the exterior derivative, and replace the usual derivative with the convariant derivative.


The exterior covariant derivative d :Ω p(M,E)Ω p+1(M,E)d_\nabla: \Omega^p (M, E) \to \Omega^{p + 1}(M, E) is defined by the following formula: given a pp-form ΦΩ p(M,E)\Phi \in \Omega^p (M, E), its exterior covariant derivative is given by

(d Φ)(X 0,,X p)= i=0 p(1) i X iΦ(X 0,,X i^,,X p) + i<j(1) i+jΦ([X i,X j],X 0,,X i^,,X j^,,X p), \begin{aligned} (\mathrm{d}_\nabla \Phi)(X_0, \dots, X_p) =& \displaystyle\sum_{i = 0}^p (-1)^i \nabla_{X_i}\Phi (X_0, \dots, \hat{X_i}, \dots, X_p) \\ &+ \displaystyle\sum_{i \lt j} (-1)^{i + j} \Phi([X_i, X_j], X_0, \dots, \hat{X_i}, \dots, \hat{X_j}, \dots, X_p), \end{aligned}

where each X iX_i is a vector field on MM, and X^\hat{X} means omission of XX.

In the case of the trivial bundle with the trivial connection, this gives the usual exterior derivative of a vector-valued differential form.

Definition on principal GG-bundles

Let p:PMp: P \to M be a principal GG-bundle. Let ωΩ 1(P,𝔤)\omega \in \Omega^1(P,\mathfrak{g}) be a connection on PP. Let H:TPTPH: T P \to T P be the horizontal projection given by ω\omega.

We let VV be a vector space, and ρ:GGL(V)\rho: G \to \GL(V) be a representation of GG on VV.


A differential form ψΩ k(P,V)\psi \in \Omega^k(P,V) is called:

  • horizontal, if ψ q(v 1,,v k)=0\psi_q(v_1,\dots,v_k)=0 whenever one of the vectors v iv_i is vertical.

  • equivariant, if r g *ψ=ρ(g 1,ψ)r_g^{*}\psi = \rho(g^{-1},\psi) for all gGg\in G, where r g:PPr_g: P \to P denotes the right action of GG on PP.

We denote by Ω ρ k(P,V)\Omega^k_{\rho}(P,V) the space of horizontal and equivariant forms. Note that Ω ρ(P,V)\Omega_{\rho}(P,V) is in general not closed under the ordinary exterior derivative. There is a canonical isomorphism

Ω ρ k(P,V)Ω k(M,P× ρV), \Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V),

where P× ρVP \times_{\rho}V is the vector bundle associated to PP via the representation ρ\rho.


The exterior covariant derivative for forms on PP

d ω:Ω k(P,V)Ω k+1(P,V) \mathrm{d}_{\omega}: \Omega^k(P,V) \to \Omega^{k+1}(P,V)

is defined by

(d ωψ) q(v 1,,v k):=(dφ) q(Hv 1,,Hv k). (\mathrm{d}_{\omega}\psi)_q(v_1,\dots,v_k) := (\mathrm{d}\varphi)_q(H v_1,\dots,H v_k).

Every form in the image of d ω\mathrm{d}_{\omega} is horizontal. If a form ψ\psi is equivariant, d ωψ\mathrm{d}_{\omega} \psi is also equivariant.


The restriction of d ω\mathrm{d}_{\omega} to Ω ρ k(P,V)\Omega^k_{\rho}(P,V) can be described in terms of the connection 1-form ωΩ 1(P,𝔤)\omega\in \Omega^1(P,\mathfrak{g}) and the derivative dρ:𝔤×VV\mathrm{d}\rho: \mathfrak{g} \times V \to V of the representation ρ\rho:

d ω(ψ)=dψ+ω dρψ. \mathrm{d}_{\omega}(\psi) = \mathrm{d}\psi + \omega \wedge_{\mathrm{d}\rho} \psi\text{.}

Here we have used the following general notation: if U,V,WU,V,W are vector spaces, φΩ p(M,V)\varphi \in \Omega^p(M,V), ψΩ q(M,W)\psi\in\Omega^q(M,W) and f:V×WUf: V \times W \to U is a linear map, we have φ fψΩ p+q(M,U)\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U).


Unlike the usual exterior derivative, the exterior covariant derivative need not be nilpotent in general. Instead, we have

d ω(d ω(ψ))=Ω dρψ. \mathrm{d}_{\omega}(\mathrm{d}_{\omega}(\psi)) = \Omega \wedge_{\mathrm{d}\rho} \psi.

In particular, d ωd ω=0\mathrm{d}_{\omega} \circ \mathrm{d}_{\omega} = 0 if ω\omega is flat.


The exterior covariant derivative for forms on MM

D ω:Ω k(M,P× ρV)Ω k+1(M,P× ρV) \mathrm{D}_{\omega}: \Omega^k(M,P \times_{\rho} V) \to \Omega^{k+1}(M,P \times_{\rho}V)

is the map induced from d ω\mathrm{d}_\omega under the isomorphism Ω ρ k(P,V)Ω k(M,P× ρV)\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V).

Note that a connection on the principal bundle p:PMp: P \to M induces? a connection on the associated vector bundle P× ρVP \times_\rho V. Then the exterior covariant derivative in the sense of Definition 1 coincides with this exterior covariant derivative.


  • The connection ω\omega itself is not in Ω Ad 1(P,𝔤)\Omega_{\mathrm{Ad}}^1(P,\mathfrak{g}): it is not horizontal.

  • The curvature of ω\omega is Ω:=d ω(ω)Ω Ad 2(P,𝔤)\Omega := \mathrm{d}_\omega(\omega)\in \Omega^2_{\mathrm{Ad}}(P,\mathfrak{g}). Since d(Ad)=[,]\mathrm{d}(\mathrm{Ad})=[-,-], we have

    Ω=dω+[ωω]. \Omega = \mathrm{d}\omega + [\omega \wedge \omega]\text{.}

    The Bianchi identity is d ωΩ=0\mathrm{d}_{\omega}\Omega=0.

Under the isomorphism Ω ρ k(P,V)Ω k(M,P× ρV)\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V), the curvature Ω\Omega corresponds to a 2-form F ωΩ 2(M,Ad(P))F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P)), and the Bianchi identity corresponds to D ωF ω=0\mathrm{D}_{\omega}F_{\omega} = 0.

Revised on August 10, 2016 15:30:31 by Dexter Chua (