Pullback

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In mathematics, a pullback can be defined in several different contexts. This article focuses primarily on the pullback of tensors on differentiable manifolds.

Given a continuously differentiable map from one differentiable manifold to another, there is an associated mapping from the cotangent bundle of Y to that of X, known as the pullback, and frequently denoted by f*. More generally, any covariant tensor pulls back under f*.

When the map f is a diffeomorphism of a manifold to itself, 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.

Contents 1 Pullback on tensors 2 Pullback of (co)tangent bundles 3 Pullback on tensor bundles 4 Pullback of diffeomorphisms 5 See also 6 References

[edit] Pullback on tensors

Let be a linear map between vector spaces V and W. Then given a tensor T of rank (0,n) on W, another tensor, the pullback f ^* T on V can be defined by precomposing with f. That is, given a tensor

and a set of vectors

one then defines the pullback as

The result f ^* T is again a tensor, so that f ^* 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 , the dual space of W, then , and so the pullback acts in the reversed direction:

.

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 indices. 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 is invertible. The pullback of a tensor of rank (n,0) can be defined by employing ; we make use of the identity (f ^{− 1}) ^* = (f ^* ) ^{− 1}. One then defines

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.

[edit] Pullback of (co)tangent bundles

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 π_M and π_N are the natural projections. Perhaps the easiest way to understand the pullback is in terms of the pushforward of f. Picking a point , the pushforward at p is a linear map between the tangent spaces

where L(V,W) being the set of linear mappings from the vector space V = T_pM to the vector space W = T_f(p)N. 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 . Thus, if we take

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

.

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,

f ^* (p) = [f _* (p)]^T.

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

.

[edit] 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 at point p in M, one defines the tensor space at point p as the n-fold tensor product

.

The pullback then proceeds analogously to the tensor space defined through an n-fold tensor product of . This definition applies as well to the exterior bundles Λ^kT*N and Λ^kT*M, which 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

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

f ^* (dω) = d(f ^* ω)

for ω in Λ(M).

[edit] 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

can be inverted to define

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

.

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.

[edit] See also

• Pullback bundle
• Pullback (category theory)

[edit] 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.