- 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 diagram commutes:
Here T*M and T*N are the cotangent bundles of M and N respectively, and πM and πN are the natural projections. (The article on cotangent spaces provides an alternate definition of a pullback, anchored in the context differential forms).
More generally, one can construct the pullback map between the exterior bundles ΛkT*N and ΛkT*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.
When M = N, then the pullback and the pushforward describe the transformation properties of a tensor on the manifold M. In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.
In category theory, 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).
See also
References
- Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See sections 1.5 and 1.6.
- Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.7 and 2.3.