Pullback - Definition and Overview

This article discusses the pullback in differential geometry. For the pullback in category theory see pullback (category theory).

In mathematics, the pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : T*N → T*M, for which the following digram commutes:

Here T*M and T*N are the cotangent bundles of M and N respectively, and π_M and π_N are the natural projections. The article on cotangent spaces has the precise definition.

More generally, one can construct the pullback map between the exterior bundles Λ^kT*N and Λ^kT*M. The pullback map is such that it maps smooth sections to smooth sections. That is, the pullback of a differential form on N is a differential form on M.

The pullback map gives rise to a contravariant functor from the category of smooth manifolds to the category of smooth vector bundles via the maps M ↦ T*M and (f : M → N) ↦ (f* : T*N → T*M).