nLab 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




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\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.

Notions of pullback:


A standard reference is

  • Bott, Tu, Differential forms in algebraic topology.

See also for instance

Last revised on December 2, 2020 at 17:31:52. See the history of this page for a list of all contributions to it.