pullback of a differential form


Differential geometry

differential geometry

synthetic differential geometry








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

If differential forms are defined by Yoneda extension from differential forms on Cartesian spaces then pullback is given on X= k˜X = \mathbb{R}^{\tilde k} and Y= kY = \mathbb{R}^k and on 1-forms

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

by the rule

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

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 constritutes 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

Revised on April 30, 2017 14:11:16 by Gabriel Chicas Reyes? (