12.16.2 Tensor Pullback Target Tensor

Tensor Pullback Target Tensor is the covariant tensor T defined on the codomain W of the pullback source map f : V → W, serving as the input object that the pullback operation f* acts upon to produce a corresponding tensor on V. Where the source map fixes the direction and endpoints of the transport, the target tensor is the specific piece of data being transported: it lives entirely on the target space W, prior to and independent of any pullback, and the pullback operation's entire purpose is to manufacture, from T and f together, a new tensor f*T living on V.

Identity of the Target Tensor

Defined Purely on the Codomain

The target tensor T is a covariant k-tensor belonging to the space of such tensors on W, denoted T⁰ₖ(W). Its definition — how it evaluates on tuples of vectors from W — makes no reference whatsoever to V or to f; these enter the picture only once the pullback operation is applied to T.

Distinguishing the Target Tensor from the Resulting Pullback

The target tensor T is not the same object as its pullback f*T: T takes arguments from W, while f*T takes arguments from V. Referring to "the tensor" without specifying whether the target tensor on W or its pullback on V is meant leaves the reference ambiguous, since the two objects generally have different domains and, except in trivial cases, different numerical values even when compared through f.

Requirements on the Target Tensor for the Pullback to Apply

Covariance Is Essential

The pullback formula requires substituting f(vᵢ) for each argument of T, which is only sensible if T accepts elements of W as inputs — precisely what it means for T to be covariant (rank (0,k)). A contravariant tensor, which instead produces elements of W as part of its data, cannot serve as a target tensor for the ordinary pullback construction without additional assumptions such as invertibility of f.

No Constraint on Symmetry or Rank Beyond Covariance

The target tensor may be of any covariant rank k ≥ 0, symmetric, antisymmetric, or neither; the pullback formula (f*T)(v₁,...,vₖ) = T(f(v₁),...,f(vₖ)) applies uniformly regardless of these additional properties. Any special structure the target tensor carries, such as antisymmetry, is automatically inherited by the pullback, as shown by direct substitution into the defining formula.

Diagram of the Target Tensor Prior to Pullback

The Target Tensor's Role Across Repeated Pullbacks

Reusability Across Different Source Maps

Because the target tensor T is defined independently of any particular source map, the same T can be pulled back along several different maps f₁ : V₁ → W, f₂ : V₂ → W, producing distinct pullback tensors f₁*T and f₂*T on unrelated spaces V₁ and V₂. The target tensor's own definition never needs to change to support this reuse.

Fixed Point Under the Identity Source Map

When the source map is the identity on W itself, the pullback of the target tensor returns the target tensor unchanged, confirming that the pullback construction, applied to the trivial source map, does nothing more than hand T back to itself, consistent with T being the fixed starting data of the whole operation.

Composability of Repeated Pullbacks on the Same Target Tensor

If the target tensor T on W is first pulled back along g : U → W and the result is then pulled back again along f : V → U, the outcome equals pulling T back directly along the single composite map g ∘ f. The target tensor itself never changes identity throughout this process; only the accumulated source map applied to it changes.

Consequences of Choosing a Different Target Tensor

Linearity in the Choice of Target Tensor

Since the pullback operation is linear in its tensor argument, replacing the target tensor T with a linear combination aT + bS of two target tensors on W produces the pullback a(f*T) + b(f*S), so the effect of varying the target tensor while holding the source map fixed is itself entirely linear and predictable from the pullbacks of the individual pieces.

Distinct Target Tensors Generally Yield Distinct Pullbacks

If f is surjective, distinct target tensors T ≠ S on W always produce distinct pullbacks f*T ≠ f*S on V, since any discrepancy between T and S on some tuple in W can be witnessed by a tuple in V mapping onto it. When f is not surjective, however, T and S may differ only on values f never attains, in which case their pullbacks can coincide despite T and S themselves being different target tensors.