nLab
pullback of a differential form

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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{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:XYf \colon X \to Y a smooth function between smooth manifold, and for ωΩ n(Y)\omega \in \Omega^n(Y) a differential n-form, there is the pullback nn form f *ωΩ n(X)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 *ω(v 1,,v n)=ω(f *v 1,,f *v n). 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= k˜X = \mathbb{R}^{\tilde k} and Y= kY = \mathbb{R}^k Cartesian spaces and f:XYf \;\colon\; X \longrightarrow Y a smooth function between them, and on differential 1-forms

ω= i=1 kω idx i \omega = \sum_{i = 1}^k \omega_i \mathbf{d}x^i

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

f *dx i j=1 k˜f ix˜ kdx˜ 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 *ω=f *( iω idx i) i=1 k(f *ω) i j=1 k˜f ix˜ jdx˜ j, 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 *ω if^* \omega_i is the pullback of functions defined by

    (f *ω i)(x)=ω i(f(x))xX (f^* \omega_i)(x) = \omega_i(f(x)) \;\;\;\forall x \in X
  • the function

    f ix˜ j: k˜ \frac{\partial f^i}{\partial \tilde x^j} \colon \mathbb{R}^{\tilde k} \to \mathbb{R}

    is the partial derivative of the kk-th coordinate component of ff along the jjthe 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 *d Y=d Xf *. f^* \circ \mathbf{d}_Y = \mathbf{d}_X \circ f^* \,.

Hence it constitutes a chain map between the de Rham complexes

f *:Ω (Y)Ω (X). 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.

See also for instance section 2.7 of

Revised on September 22, 2017 02:45:28 by Urs Schreiber (77.56.177.247)