nLab
pullback of a differential form

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

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)

          Contents

          Definition

          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 \,,

          where

          • 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.

          Properties

          Compatibility with the de Rham differential

          Proposition

          (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.

          References

          A standard reference is

          • Bott, Tu, Differential forms in algebraic topology.

          See also for instance section 2.7 of

          Last revised on June 19, 2018 at 04:22:55. See the history of this page for a list of all contributions to it.