# nLab pullback of a differential form

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

For $f:X\to Y$ a smooth function between smooth manifold, and for $\omega \in {\Omega }^{n}\left(Y\right)$ a differential n-form, there is the pullback $n$ form ${f}^{*}\omega \in {\Omega }^{n}\left(X\right)$.

### 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?

${f}^{*}\omega \left({v}_{1},\cdots ,{v}_{n}\right)=\omega \left({f}_{*}{v}_{1},\cdots ,{f}_{*}{v}_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $X={ℝ}^{\stackrel{˜}{k}}$ and $Y={ℝ}^{k}$ and on 1-forms

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

by the rule

${f}^{*}d{x}^{i}≔\sum _{j=1}^{\stackrel{˜}{k}}\frac{\partial {f}^{i}}{\partial {\stackrel{˜}{x}}^{k}}d{\stackrel{˜}{x}}^{j}$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

${f}^{*}\omega ={f}^{*}\left(\sum _{i}{\omega }_{i}d{x}^{i}\right)≔\sum _{i=1}^{k}{\left({f}^{*}\omega \right)}_{i}\sum _{j=1}^{\stackrel{˜}{k}}\frac{\partial {f}^{i}}{\partial {\stackrel{˜}{x}}^{j}}d{\stackrel{˜}{x}}^{j}\phantom{\rule{thinmathspace}{0ex}},$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

$\left({f}^{*}{\omega }_{i}\right)\left(x\right)={\omega }_{i}\left(f\left(x\right)\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\forall x\in X$(f^* \omega_i)(x) = \omega_i(f(x)) \;\;\;\forall x \in X
• the function

$\frac{\partial {f}^{i}}{\partial {\stackrel{˜}{x}}^{j}}:{ℝ}^{\stackrel{˜}{k}}\to ℝ$\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:

${f}^{*}\circ {d}_{Y}={d}_{X}\circ {f}^{*}\phantom{\rule{thinmathspace}{0ex}}.$f^* \circ \mathbf{d}_Y = \mathbf{d}_X \circ f^* \,.

Hence it constritutes a chain map between the de Rham complexes

${f}^{*}:{\Omega }^{•}\left(Y\right)\to {\Omega }^{•}\left(X\right)$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.