12.17.3 Tensor Pushforward Vector Slot Action

Tensor Pushforward Vector Slot Action is the rule describing how each contravariant argument slot of a tensor transforms under the pushforward operation, namely that a vector occupying such a slot is replaced by its image under the pushforward's source map f : V → W, applied directly and in the same direction as f itself. It is the pushforward's counterpart to the pullback's slot transformation, differing in one essential respect: because a contravariant slot holds an output vector rather than an input argument, the substitution moves forward along f rather than requiring pre-composition, and this directional difference is the defining feature separating the two constructions.

The Forward Substitution Rule

Single-Slot Action on a Vector

For a single vector v ∈ V occupying a contravariant slot, the pushforward action replaces v with f(v) ∈ W — the ordinary image of v under f. This is the most elementary instance of the pushforward, since a vector is itself a rank-1 contravariant tensor, and pushing it forward is nothing more than applying f directly.

Action Across the Slots of a Simple Contravariant Tensor

For a simple contravariant k-tensor built as a tensor product of k vectors, the pushforward acts on each factor independently by applying f to it, exactly mirroring the way the pullback's slot transformation acts independently on each covariant argument, but with the substitution running in the forward direction of f rather than requiring the tensor to be pre-composed with f.

Extension to General Contravariant Tensors by Linearity

For a general contravariant k-tensor expressed as a finite sum of simple tensors, T = Σ v₁ⁱ ⊗ ... ⊗ vₖⁱ, the pushforward is obtained by applying the vector slot action to every simple summand and then summing the results, f*T = Σ f(v₁ⁱ) ⊗ ... ⊗ f(vₖⁱ). This extension by linearity is what makes the vector slot action well-defined on the whole contravariant tensor space rather than only on simple tensors.

Diagram of the Vector Slot Action

Well-Definedness Requirements

Linearity of f Is Essential

Since each factor f(vᵢ) must respect linear combinations of vᵢ for the extension by linearity to be consistent — that is, for f((av + bv′)) = af(v) + bf(v′) to hold — the vector slot action requires f to be linear. Without this, applying the action to two different representations of the same tensor as a sum of simple tensors could produce different, inconsistent results.

No General Backward Action Without Invertibility

The vector slot action moves strictly forward along f; there is no way, using only f, to define an action moving a vector in W backward into V, unless f happens to be invertible, in which case f⁻¹ supplies the needed backward map. This is the direct counterpart of the fact that the ordinary pullback cannot act on contravariant tensors without a similar invertibility assumption.

Interaction with Rank and Symmetry

Rank Is Preserved

Exactly as with the pullback on covariant slots, the vector slot action applied across all k factors of a rank-k contravariant tensor produces another rank-k contravariant tensor, since the number of tensor factors is unchanged by applying f to each one individually.

Symmetry and Antisymmetry Are Preserved

If T is a symmetric contravariant tensor, f*T remains symmetric, since permuting the factors of T before or after applying f to each factor produces the same result, by the same slot-independence argument used for the pullback. The identical reasoning shows antisymmetric contravariant tensors — such as those used to represent oriented volumes or multivectors — remain antisymmetric under the vector slot action.

Relationship to the Pullback's Slot Transformation

Structural Mirror with Reversed Direction

The vector slot action and the pullback's slot transformation are structurally parallel: both replace the contents of each tensor slot using the source map f, and both preserve rank and symmetry class unconditionally. They differ only in which kind of slot they act on and correspondingly in which direction the induced tensor map travels — the pushforward moves contravariant tensors forward along f, from V to W, while the pullback moves covariant tensors backward along f, from W to V.

Consistency When Both Operations Apply

For a mixed tensor with both covariant and contravariant slots, when f is invertible, the two slot actions can be combined: contravariant slots are pushed forward via f, and covariant slots are pulled back via f⁻¹ acting in the pullback's role, together producing a single well-defined transport of the entire mixed tensor from V to W consistent with both the vector slot action and the pullback slot transformation applied to their respective slot types.