nLab
pullback of a differential form
Context
Differential geometry
Contents
Definition
For $f \colon X \to Y$ a smooth function between smooth manifold , and for $\omega \in \Omega^n(Y)$ a differential n-form , there is the pullback $n$ form $f^* \omega \in \Omega^n(X)$ .

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?

$f^* \omega(v_1, \cdots, v_n) = \omega(f_* v_1, \cdots, f_* v_n)
\,.$

In terms of coordinate expression
If differential forms are defines by Yoneda extension from differential forms on Cartesian spaces then pullback is given on $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^k$ and on 1-forms

$\omega = \sum_{i = 1}^k \omega_i \mathbf{d}x^i$

by the rule

$f^* \mathbf{d}x^i
\coloneqq
\sum_{j = 1}^{\tilde k} \frac{\partial f^i}{\partial \tilde x^k} \mathbf{d}\tilde x^j$

and hence

$f^* \omega = f^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right)
\coloneqq
\sum_{i = 1}^k \left(f^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial f^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j
\,,$

where

$f^* \omega_i$ is the pullback of functions defined by

$(f^* \omega_i)(x) = \omega_i(f(x)) \;\;\;\forall x \in X$

the function

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

is the partial derivative of the $k$ -th coordinate component of $f$ along the $j$ the coordinate.

Properties
Compatibility with the de Rham differential
Pullback of differential forms commutes with the de Rham differential :

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

Hence it constritutes a chain map between the de Rham complexes

$f^* \colon \Omega^\bullet(Y) \to \Omega^\bullet(X)$

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

Revised on February 5, 2013 20:42:59
by

Urs Schreiber
(131.174.41.0)