3.16.2 Tensor Covector Pullback Input

Tensor Covector Pullback Input is the covector supplied to the pullback operation as the object being transported from the codomain of a linear map back to its domain. In the pullback formula (f*φ)(v) = φ(f(v)), associated with a linear map f: V → W, the pullback input is φ ∈ W*, the covector defined on W that is fed into the operation, distinct from the source map f that supplies the route and from the resulting pullback covector f*φ that is produced on V. Understanding the pullback input as a distinct component of the operation clarifies what varies, what is held fixed, and what constraints the input must satisfy for the pullback to be defined.

Identifying the Pullback Input

Where the Input Sits in the Formula

The pullback operation f*: W* → V* takes as input a single covector φ belonging to W*, the dual space of the codomain of f. This is the only input the operation requires beyond the fixed source map f; no vector from V or W is supplied directly, since the operation f* itself is a map between spaces of covectors, and its output f*φ is only evaluated on vectors of V after the pullback has already been formed.

The Input's Domain of Definition

The pullback input φ must be a linear functional on W, the codomain of f, meaning φ: W → F where F is the underlying field. If a functional is defined on a different space, such as V itself or some unrelated space, it is not a valid input to the pullback operation associated with f, since the defining formula φ(f(v)) requires φ to accept f(v) ∈ W as its argument.

Varying the Input While Fixing the Source Map

Linearity in the Input

For a fixed source map f, the pullback operation is linear in its input: for covectors φ, ψ ∈ W* and scalars a, b,

This linearity means the pullback input can be varied freely across all of W*, and the output tracks that variation linearly, which is exactly the statement that f* is a linear map with domain W*.

Basis Elements as Canonical Inputs

If ᵤ^1, ..., ᵤ^m is a dual basis for W*, feeding each basis covector ᵤ^i individually into the pullback operation and recording the results f*ᵤ^1, ..., f*ᵤ^m completely determines the pullback operation, since every other input is a linear combination of these basis covectors and the pullback respects linear combinations.

Constraints and Degenerate Inputs

The Zero Covector as Input

Feeding the zero covector 0 ∈ W* into the pullback operation always produces the zero covector on V, since (f*0)(v) = 0(f(v)) = 0 for every v. This holds regardless of the choice of source map f, making the zero input the unique element of W* guaranteed to produce a trivial output for every possible f.

Inputs Vanishing on the Image of f

A nonzero pullback input φ can still produce a zero output f*φ = 0 if φ vanishes on the entire image of f, that is, if φ(f(v)) = 0 for all v ∈ V. Such inputs form the annihilator of the image of f within W*, and their presence shows that the pullback operation is not injective on its input whenever f fails to be surjective.

The Input's Role in Extending Pullback to Tensors

Multiple Inputs for Higher-Order Tensors

When the pullback operation is extended from single covectors to covariant tensors of type (0, q), the input becomes a q-linear functional T: W × ... × W → F rather than a single covector, but the same principle applies: the pullback f*T is defined by routing all q arguments through f before evaluating T, so the pullback input in this generalized sense is the entire multilinear object T, not a single covector.

Consistency Across Input Types

Whether the pullback input is a single covector or a higher covariant tensor, it must always be a purely covariant object valued on W, since only such objects can accept the images f(v_1), ..., f(v_q) as arguments. This consistency is what allows the pullback operation to be described uniformly across all covariant tensor orders using the same defining principle applied to whatever input is supplied.

Diagrammatic Summary

The diagram treats the pullback operation as a box labeled f*, with the covector φ entering as its input from the left and the resulting covector f*φ exiting as its output on the right, emphasizing the input's role as the variable quantity supplied to a fixed operation.