# nLab pullback of a differential form

### Context

#### Differential geometry

differential geometry

synthetic 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

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

### 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.