pullback of a differential form



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




  • (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 }



          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)



          For f:XYf \colon X \to Y a smooth function between smooth manifold, and for ωΩ n(Y)\omega \in \Omega^n(Y) a differential n-form, there is the pullback nn-form f *ωΩ n(X)f^* \omega \in \Omega^n(X).

          In terms of push-forward of vector fields

          If differential forms are defined as linear duals to vectors then pullback is the dual operation to pushforward of a vector field?

          f *ω(v 1,,v n)=ω(f *v 1,,f *v n). f^* \omega(v_1, \cdots, v_n) = \omega(f_* v_1, \cdots, f_* v_n) \,.

          In terms of coordinate expression

          Differential forms may be defined by Yoneda extension from differential forms on Cartesian spaces (see at geometry of physics – differential forms).

          For X= k˜X = \mathbb{R}^{\tilde k} and Y= kY = \mathbb{R}^k Cartesian spaces and f:XYf \;\colon\; X \longrightarrow Y a smooth function between them, and on differential 1-forms

          ω= i=1 kω idx i \omega = \sum_{i = 1}^k \omega_i \mathbf{d}x^i

          the pullback operation f *f^\ast is given by

          f *dx i j=1 k˜f ix˜ jdx˜ j f^* \mathbf{d}x^i \;\coloneqq\; \sum_{j = 1}^{\tilde k} \frac{\partial f^i}{\partial \tilde x^j} \mathbf{d}\tilde x^j

          and hence

          f *ω=f *( iω idx i) i=1 k(f *ω) i j=1 k˜f ix˜ jdx˜ j, f^* \omega = f^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right) \coloneqq \sum_{i = 1}^k \left(f^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial f^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j \,,


          • f *ω if^* \omega_i is the pullback of functions defined by

            (f *ω i)(x)=ω i(f(x))xX (f^* \omega_i)(x) = \omega_i(f(x)) \;\;\;\forall x \in X
          • the function

            f ix˜ j: k˜ \frac{\partial f^i}{\partial \tilde x^j} \colon \mathbb{R}^{\tilde k} \to \mathbb{R}

            is the partial derivative of the kk-th coordinate component of ff along the jjthe coordinate.


          Compatibility with the de Rham differential


          (compatiblity with the de Rham differential)

          Pullback of differential forms commutes with the de Rham differential:

          f *d Y=d Xf *. f^* \circ \mathbf{d}_Y = \mathbf{d}_X \circ f^* \,.

          Hence it constitutes a chain map between the de Rham complexes

          f *:Ω (Y)Ω (X). f^* \colon \Omega^\bullet(Y) \to \Omega^\bullet(X) \,.

          Sheaf of differential forms

          Under pullback differential forms form a presheaf on the catories CartSp and SmthMfd, in fact a sheaf with respect to the standard open cover-coverage.


          A standard reference is

          • Bott, Tu, Differential forms in algebraic topology.

          See also for instance section 2.7 of

          Last revised on November 18, 2018 at 06:23:37. See the history of this page for a list of all contributions to it.