12.16 Tensor Object Pullback Operation
The Tensor Object Pullback Operation transforms tensors through mappings, preserving structure while adapting to new spaces in algebraic geometry and differential calculus.
Tensor Object Pullback Operation is the operation that transports a covariant tensor defined on one vector space back to a covariant tensor on another vector space, using a fixed linear map between the two spaces to pre-compose the tensor's arguments, formally written f* : T⁰ₖ(W) → T⁰ₖ(V) for a linear map f : V → W acting on rank-k covariant tensors. It is called a "pullback" because it moves tensors in the direction opposite to the map f itself — tensors are drawn backward from the codomain W into the domain V — and it is the natural counterpart in tensor algebra to pushing vectors forward along f, applicable specifically to covariant objects such as bilinear forms, differential forms, and general multilinear functionals rather than to vectors or contravariant tensors.
Defining Formula
Pullback of a Covariant k-Tensor
Given a covariant k-tensor T on W and vectors v₁, ..., vₖ ∈ V, the pullback f*T is defined by first mapping each vector into W via f, then evaluating T on the resulting tuple. This single formula covers every covariant rank: setting k = 1 recovers the pullback of a linear functional, and k = 2 recovers the pullback of a bilinear form.
Well-Definedness as a Multilinear Object
Because f is linear and T is multilinear, the composite assignment (v₁, ..., vₖ) ↦ T(f(v₁), ..., f(vₖ)) is itself multilinear in v₁, ..., vₖ, confirming f*T is a genuine covariant k-tensor on V and not merely some unstructured function of k vector arguments.
Basic Properties of the Pullback Operation
Linearity in the Tensor Argument
For a fixed source map f, the pullback operation is linear in the tensor being pulled back, following directly from the fact that evaluation of a linear combination of tensors at a fixed tuple of vectors equals the corresponding linear combination of the individual evaluations.
Contravariant Composition Law
Pulling back along a composite map g ∘ f equals pulling back along g first and then along f, reversing the order of composition. This is what makes the pullback operation contravariant: it turns a chain of forward maps into the corresponding chain of pullbacks in exactly the opposite order.
Compatibility with the Tensor Product
The pullback of a tensor product of two covariant tensors equals the tensor product of the individual pullbacks, since both sides, evaluated on a combined tuple of vectors, reduce to applying f to each vector once and evaluating T and S on the appropriate sub-tuples.
Diagram of the Pullback's Reversed Direction
Contrast with Pushforward and with Contravariant Tensors
Pullback Applies to Covariant Objects, Not Vectors
A vector v ∈ V is pushed forward by f to f(v) ∈ W, moving in the same direction as f; it is not pulled back. The pullback operation specifically applies to covariant tensors — objects that themselves consume vectors as arguments — since only such objects can be meaningfully pre-composed with f in the manner the defining formula requires.
No General Pullback for Mixed or Contravariant Tensors
A general linear map f does not induce a pullback on contravariant tensors (such as vectors themselves) or on mixed tensors with contravariant slots, because there is no way, using only f and no additional structure such as an inverse, to move a contravariant index backward against f's own direction. Pullback of contravariant or mixed tensors typically requires f to be invertible, using f⁻¹ for the contravariant slots.
Relationship to the Pushforward When f Is Invertible
When f is a linear isomorphism, both a pushforward (using f) and a pullback (using f⁻¹ in place of f in the defining formula) are available on covariant tensors, and the two operations are inverse to one another: pushing forward and then pulling back along the same isomorphism returns the original tensor unchanged.
Typical Uses of the Pullback Operation
Transporting Metrics and Bilinear Forms
Given an inner product or general bilinear form β on W, its pullback along f produces a bilinear form on V, allowing geometric or algebraic structure defined on W to be studied indirectly on V through the lens of the map f, without requiring f to be invertible or even injective.
Restriction to a Subspace via Inclusion
When f is specifically an inclusion map ι : U → T embedding a subspace, the pullback ι* restricts a covariant tensor defined on the ambient space T down to the subspace U, giving the tensor's "values as seen from within U" — a common and structurally simple special case of the general pullback operation.
Change of Coordinates
When f represents a linear change-of-coordinates map between two coordinate systems, pulling back a tensor expressed in one coordinate system produces its coordinate expression in the other, making the pullback operation the formal underpinning of how tensor components are converted between different bases or coordinate charts.