From Academic Kids

In mathematics, a pullback can be defined in several related contexts. Given a function on vector spaces, a pullback can be defined on the space of tensors. Given a smooth map between differentiable manifolds, a pullback can be defined on the cotangent bundle. When the smooth map is a self-diffeomorphism, then the pullback, together with the pushforward, describe the transformation properties of the manifold under a change of coordinates. Using traditional language, these describe the transformation properties of contravariant and covariant tensors. These definitions are all given below. In addition, a pullback may be defined in category theory; this is presented in the article pullback (category theory).


Pullback on tensors

Let <math>f:V\rightarrow W<math> be a linear map <math>f\in L(V,W)<math> between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback <math>f^*T<math> on V can be defined. That is, given a tensor

<math>T:W \times W \times ... \times W \rightarrow \mathbb{R}<math>

and a set of vectors

<math>(v_1,v_2,...,v_n) \in V \times V\times ...\times V<math>

one then defines the pullback as

<math>(f^*T)(v_1,v_2,...,v_n) = T(f(v_1), f(v_2), ... ,f(v_n))<math>.

The result <math>f^*T<math> is again a tensor, so that <math>f^*<math> is in fact a mapping from tensors on W to tensors on V. As a special case, note that if T is a (0,1)-tensor, so that <math>T\in W^*<math>, the dual space of W, then <math>f^*T\in V^*<math>, and so the pullback acts in the reversed direction:

<math>f^*:W^*\rightarrow V^*<math>.

For a general f, a pullback can only be defined on tensors of rank (0,n). This is precisely because a pullback on mixed tensors would need to be "going in the opposed direction" for the contravariant indeces. If f is invertible, then this can be done, and one can define the pullback of an arbitrary mixed-rank tensor, as shown next. Let f be a linear isomorphism, so that <math>f \in GL(V,W)<math> is invertible. The pullback of a tensor of rank (n,0) can be defined by employing <math>(f^{-1})^*:V^* \rightarrow W^*<math>; we make use of the identity <math>(f^{-1})^*=(f^*)^{-1}<math>. One then defines

<math>(f^*T)(v^*_1,v^*_2,...,v^*_n) =

T((f^{-1})^*(v^*_1), (f^{-1})^*(v^*_2), ... ,(f^{-1})^*(v^*_n))<math>

Thus we've shown that if a linear transformation is invertible, it can be used to define the pullback on general tensors of mixed rank (m,n). Perhaps the easiest way to visualize and understand the above is to keep firmly in mind that f is nothing more than a matrix, so that f(v) is just the multiplication of a vector by a matrix. Similarly, the dual space should be visualized as nothing more than a dot product.

Pullback on manifolds

The pullback of smooth map f : MN between differentiable manifolds is a smooth vector bundle morphism f* : T*NT*M, for which the following diagram commutes:

Missing image

Here T*M and T*N are the cotangent bundles of M and N respectively, and πM and πN are the natural projections. Perhaps the easiest way to understand the pullback is in terms of the pushforward of f. Picking a point <math>p\in M<math>, the pushforward at p is a linear map between the tangent spaces

<math>f_*(p) \in L(T_pM,T_{f(p)}N)<math>

where L(V,W) being the set of linear mappings from the vector space <math>V = T_pM<math> to the vector space <math>W = T_{f(p)}N<math>. The cotangent space is dual to the tangent space, and maps on the dual space act as the transpose. That is, consider two ordinary vectors v and w, and a matrix A. The dot product obeys the identity <math>w\cdot Av = (A^Tw)\cdot v<math>. Thus, if we take

<math>A=f_*(p)\in L(T_pM,T_{f(p)}N)<math>

then the transpose is going to act on the 1-forms:

<math>A^T=[f_*(p)]^T\in L(T^*_{f(p)}N,T^*_pM )<math>.

We use the transpose to map the cotangent spaces. For each point in the manifold, the pullback is defined as the matrix transpose of the pushforward; that is,

<math>f^*(p)= [f_*(p)]^T<math>.

Note that this mapping is in a certain sense going in the "backwards" direction, that is,

<math>f^*:T^*N \rightarrow T^*M<math>.

Pullback on tensor bundles

More generally, one can construct the pullback map between tensor bundles of rank (0,n); the construction proceeds entirely analogously to that for a tensor. That is, by considering the cotangent space <math>V^*=T^*_pM<math> at point p in M, one defines the tensor space at point p as the n-fold tensor product

<math>V^* \otimes V^* \otimes ... \otimes V^*<math>.

The pullback then proceeds analogously to the tensor space defined through an n-fold tensor product of <math>W^*=T^*_{f(p)}N<math>. This definition applies as well to the exterior bundles ΛkT*N and ΛkT*M, are strict subspaces of the general tensor bundles, closed under the exterior algebra. The pullback operation commutes with the exterior algebra, and so the pullback of an alternating form is again an alternating form. That is, the pullback of a differential form on N is a differential form on M. Symbolically, we write

<math>f^*(\alpha \wedge \beta)=f^*\alpha \wedge f^*\beta<math>

for α and β in Λ(M). Similarly, the pullback is natural with respect to derivations:

<math>f^*(d\omega) = d(f^*\omega)<math>

for ω in Λ(M).

Pullback of diffeomorphisms

When the map f between manifolds is a diffeomorphism, that is, it is both smooth and invertible, then the pullback can be defined for the tangent space as well as for the cotangent space, and thus, by extension, for an arbitrary mixed tensor bundle on the manifold. The matrix

<math>A=f_*(p)\in GL(T_pM,T_{f(p)}N)<math>

can be inverted to define

<math>A^{-1}=[f_*(p)]^{-1} \in GL(T_{f(p)}N, T_pM)<math>

and thus one has, at each point p, that the pushforward is the inverse of the pullback, now acting on the tangent space (instead of the cotangent space):


so that

<math>f^*:TN\rightarrow TM<math>.

A general mixed tensor will then transform as a mixture of transposes and inverses, depending on whether the indices are contra- or co-variant. 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.

Pullback in category theory

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 MT*M and (f : MN) ↦ (f* : T*NT*M). See the article pullback (category theory) for details.


  • 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.

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