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Exterior covariant derivative

We require:

  • A smooth manifold MM.

  • A Lie group GG with Lie algebra 𝔤\mathfrak{g}.

  • A principal GG-bundle PP over MM.

  • A finite-dimensional representation ρ:GV×V\rho: G \to V \times V.

Definition 1. A 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,...,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 k(P,V)\Omega_{\rho}^k(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.

Definition 2. Let ωΩ 1(P,𝔤)\omega \in \Omega^1(P,\mathfrak{g}) be a connection on PP. 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,...,v_k) := (\mathrm{d}\varphi)_q(Hv_1,...,Hv_k)\text{,}

where H:TPTPH: TP \to TP denotes the projection to the horizontal subspace defined by ω\omega.

Some facts are:

  • Every form in the image of d ω\mathrm{d}^{\omega} is horizontal. If a form ψ\psi is equivariant, d ω\mathrm{d}^{\omega} 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).

  • The previous fact can be used to show

    d ω(d ω(ψ))=Ω dρψ, \mathrm{d}^{\omega}(\mathrm{d}^{\omega}(\psi)) = \Omega \wedge_{\mathrm{d}\rho} \psi\text{,}

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

Definition 3. 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).

Examples.

  • 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 15, 2010 19:59:29 by Toby Bartels (75.88.65.54)