12.17.1 Tensor Pushforward Source Map

The Tensor Pushforward Source Map describes how tensors transform under mappings, linking source and target spaces through linear operations.

Tensor Pushforward Source Map is the underlying linear map f : V → W that a tensor pushforward operation uses to transport contravariant tensors — vectors and their tensor products — forward from V into W, in the same direction that f itself points. It plays the same foundational role for the pushforward operation that the pullback source map plays for the pullback: it is the single piece of fixed data that must be specified before "the pushforward of a tensor" has any meaning, and every subsequent property of the pushforward operation is inherited directly from properties of this map.

Role of the Source Map in Defining the Pushforward

Fixing the Forward Direction of Transport

f : V ​ ​ \to ​ ​ W ​ ​, ​ ​ f_{*} : T_{k} (V) ​ ​ \to ​ ​ T_{k} (W)

The pushforward f* moves in the same direction as the source map: contravariant tensors on V are sent to contravariant tensors on W, matching the direction f itself already points. This is the key structural difference from the pullback source map, whose induced operation runs in the opposite direction from f.

Necessity of the Source Map's Own Linearity

Because the pushforward is extended from simple tensors to general contravariant tensors by linearity — summing the pushforwards of each simple summand — the source map f must itself be linear for this extension to be well-defined and independent of how a given tensor happens to be written as a sum of simple tensors.

Formula for the Pushforward in Terms of the Source Map

Pushforward of a Vector

f_{*} (v) = f (v)

For the simplest case, a rank-1 contravariant tensor (an ordinary vector), the pushforward is exactly the source map applied directly, with no further modification: the pushforward of a vector is literally its image under f.

Pushforward of a Simple Higher-Rank Contravariant Tensor

f_{*} (v_{1} \otimes \dots \otimes v_{k}) = f (v_{1}) \otimes \dots \otimes f (v_{k})

The source map appears once per tensor factor of a simple contravariant k-tensor, applied identically and independently to each factor. This uniform per-factor appearance is exactly the vector slot action, and it is entirely determined by which map f has been designated as the pushforward's source map.

Diagram of the Source Map and Its Forward Pushforward

Consequences of Properties of the Source Map

Injectivity Preserves Distinctness of Simple Tensors

If the source map f is injective, distinct simple contravariant tensors built from linearly independent vectors in V remain distinguishable after pushforward, since f sends linearly independent tuples to linearly independent tuples in this case. Without injectivity, distinct tensors on V can collapse to the same pushforward tensor on W.

Covariant Composition Law

(g \circ f)_{*} = g_{*} \circ f_{*}

If two source maps f : V → W and g : W → U are composed to give g ∘ f : V → U, the pushforward along the composite equals the pushforward along g applied after the pushforward along f, in the same order as the maps themselves compose. This is the covariant analogue of the pullback's contravariant composition law, distinguished by preserving rather than reversing the order of composition, exactly because the pushforward moves with f rather than against it.

Identity Source Map Yields the Identity Pushforward

When the source map is the identity id_V : V → V, the pushforward (id_V)* acts as the identity on every contravariant tensor, since applying id_V to each vector factor leaves it unchanged. This anchors the covariant composition law in its simplest case and mirrors the corresponding fact for the pullback's identity source map.

Distinguishing the Source Map from the Tensor Being Pushed Forward

Two Independent Pieces of Data

Every pushforward computation involves the fixed source map f, used repeatedly, and the specific contravariant tensor on V being transported, which may vary between applications while f stays the same. As with the pullback, omitting which source map was used when describing "the pushforward" of a tensor leaves the description incomplete, since different choices of f generally send the same tensor to different results in W.

Shared Terminology, Opposite Direction from the Pullback

Both the pullback and the pushforward are described relative to a source map with the same name and the same underlying linear map f, but the pullback source map is used to transport covariant tensors backward while the pushforward source map is used to transport contravariant tensors forward; distinguishing which of the two operations is intended is essential whenever both covariant and contravariant tensors are present in the same discussion.