nLab pullback of a differential form

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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 $f^* \omega \in \Omega^n(X)$ .

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) \,.

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.

(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) \,.

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.

Notions of pullback:

A standard reference is