12.16.3 Tensor Pullback Slot Transformation

Tensor Pullback Slot Transformation is the precise rule describing how each individual argument slot of a covariant tensor is transformed when the pullback operation is applied, namely that every one of the tensor's k argument slots independently receives the same substitution — replacing a vector v from the source space V with its image f(v) in the target space W — before the original tensor is evaluated. Where the pullback operation as a whole describes the transport of an entire tensor, the slot transformation isolates the mechanism operating uniformly at the level of a single argument position, showing that the pullback's global effect is nothing more than this one local substitution repeated once per slot.

The Uniform Substitution Rule

Single-Slot Transformation

Every argument slot of a covariant tensor undergoes exactly the same transformation rule, regardless of which slot it is or what rank the tensor has: a vector supplied to that slot is first mapped through f before the underlying tensor sees it. This uniformity is what allows a single formula to describe the pullback of tensors of any covariant rank without needing a separate rule for each slot individually.

Assembling the Slots into the Full Pullback

The full pullback formula is obtained by applying the single-slot transformation independently to each of the k arguments and then feeding the transformed tuple into the original tensor T. No interaction between different slots occurs during this transformation step; each slot is handled in isolation before the multilinear evaluation combines them.

Consequences of the Slot-by-Slot Nature

Independence from Slot Position

Because the same map f is applied identically regardless of which slot index is involved, permuting the order in which arguments are listed commutes with the slot transformation: transforming then permuting gives the same result as permuting then transforming. This is what allows the pullback operation to interact predictably with symmetrization and antisymmetrization of tensor slots.

Effect on Symmetric Tensors

If T is symmetric, applying the identical slot transformation to each argument before evaluation preserves that symmetry in the pullback, since swapping v and v′ before transformation produces the same pair of transformed arguments, in swapped order, as swapping after transformation. Antisymmetry is preserved by the identical reasoning, since the sign change under swapping is a property of T alone and is unaffected by which arguments were substituted beforehand.

Diagram of the Slot-by-Slot Substitution

Slot Transformation Under Composed Source Maps

Two-Stage Substitution Collapses to One

If the slot transformation for f is applied and then the slot transformation for g is applied to the result, the composite substitution in each slot is identical to applying the single slot transformation for the composite map g ∘ f directly. This slot-level identity is the elementary fact underlying the contravariant composition law (g∘f)* = f*∘g* for the pullback operation as a whole, since the whole-tensor identity follows immediately once it is established slot by slot.

Slot Transformation for the Identity Map

When f = id_V, the slot transformation leaves every argument unchanged, v ↦ v, so the pullback of any tensor along the identity source map reproduces the tensor's original evaluation on every tuple of arguments, matching the fact that (id_V)*T = T.

Contrast with Attempting to Transform Contravariant Slots

Why Only Covariant Slots Admit This Transformation

The slot transformation rule substitutes f(v) for arguments the tensor consumes, which is only meaningful for covariant slots that indeed accept vectors from V as valid replacements once mapped into W. A contravariant slot instead produces a vector as output rather than consuming one, so there is no argument position for the substitution v ↦ f(v) to act on, which is precisely why the ordinary pullback slot transformation does not extend to contravariant tensor slots without additional structure such as invertibility of f.