pullback of a k-form

If X is a manifoldMathworldPlanetmath, let Ωk(X) be the vector space of k-forms on X.

Definition Suppose X and Y are smooth manifolds, and suppose f is a smooth mapping f:XY. Then the pullback induced by f is the mapping f:Ωk(Y)Ωk(X) defined as follows: If ωΩk(Y), then f(ω) is the k-form on X defined by the formula


where xX, X1,,XkTx(X), and Df is the tangent map Df:TXTY.

0.0.1 Properties

Suppose X and Y are manifolds.

  • If idX is the identity map on X, then (idX) is the identity map on Ωk(X).

  • If X,Y,Z are manifolds, and f,g are mappings f:XY and g:YZ, then

  • If f is a diffeomorphism f:XY, then f is a diffeomorphism with inverse

  • If f is a mapping f:XY, and ωΩk(Y), then


    where d is the exterior derivativeMathworldPlanetmath.

  • Suppose f is a mapping f:XY, ωΩk(Y), and ηΩl(Y). Then

  • If g is a 0-form on Y, that is, g is a real valued function g:Y, and f is a mapping f:XY, then f(g)=fg.

  • Suppose U is a submanifoldMathworldPlanetmath (or an open set) in an manifold X, and ι:UX is the inclusion mapping. Then ι restricts k-forms on X to k-forms on U.

Title pullback of a k-form
Canonical name PullbackOfAKform
Date of creation 2013-03-22 14:00:34
Last modified on 2013-03-22 14:00:34
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 7
Author bwebste (988)
Entry type Definition
Classification msc 53-00
Related topic Pullback2
Related topic TangentMap