# nLab pullback of a differential form

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(ʃ \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$ʃ_{dR} \dashv \flat_{dR}$

• tangent cohesion

• differential cohomology diagram
• differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

$(\Re \dashv \Im \dashv \&)$

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

• 

\array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& &#643; &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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

Differential forms may be defined by Yoneda extension from differential forms on Cartesian spaces (see at geometry of physics – differential forms).

For $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^k$ Cartesian spaces and $f \;\colon\; X \longrightarrow Y$ a smooth function between them, and on differential 1-forms

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

the pullback operation $f^\ast$ is given by

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

and hence

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

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

###### Proposition

(compatiblity 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 constitutes 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.