nLab pullback of a differential form

Contents

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

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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.

Notions of pullback:

References

A standard reference is

• Bott, Tu, Differential forms in algebraic topology.