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exterior covariant derivative

Context

Differential geometry

synthetic differential geometry

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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

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smooth space

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cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          Contents

          Idea

          The combination of an exterior derivative with a covariant derivative.

          Definition

          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.

          Definition

          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.

          Definition

          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.

          Definition

          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).
          Proposition

          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.

          Proposition

          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).

          Proposition

          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.

          Definition

          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 coincides with this exterior covariant derivative.

          Example

          • 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.

          Applications

          Last revised on December 1, 2017 at 14:11:27. See the history of this page for a list of all contributions to it.