12.16.1 Tensor Pullback Source Map

The Tensor Pullback Source Map traces how tensor fields transform under mappings, revealing their structural dependencies and coordinate relationships.

Tensor Pullback Source Map is the underlying linear map f : V → W between vector spaces that a tensor pullback operation uses to transport covariant tensors — such as bilinear forms, differential forms, or general multilinear functionals — backward from the codomain W to the domain V, in the direction opposite to f itself. Because the pullback operation moves tensors against the arrow of f, the source map is the single piece of data that must be fixed before "the pullback of a tensor" can be assigned any meaning at all, and every property of the resulting pullback operation is inherited directly from properties of this map.

Role of the Source Map in Defining the Pullback

Fixing the Direction of Transport

f : V ​ ​ \to ​ ​ W ​ ​, ​ ​ f^{*} : T^{0} (W) ​ ​ \to ​ ​ T^{0} (V)

Given the source map f pointing from V to W, the induced pullback f* on covariant tensors points the opposite way, from tensors on W back to tensors on V. The source map is what determines both endpoints of this reversed transport: its domain V becomes the pullback's codomain, and its codomain W becomes the pullback's domain.

Necessity of the Source Map's Own Linearity

Because tensors are multilinear objects, the pullback construction requires evaluating a tensor on W against tuples of vectors obtained by applying f to vectors of V. This evaluation is only guaranteed to interact correctly with the tensor's multilinearity if f itself is linear, so the source map's linearity is a precondition for the pullback operation to be well-defined at all, not an optional convenience.

Formula for the Pullback in Terms of the Source Map

Pullback of a Bilinear Form

(f^{*} β) (v_{1}, v_{2}) = β (f (v_{1}), f (v_{2}))

Given a bilinear form β on W, its pullback along the source map f is defined by first applying f to each argument, then evaluating β. Every occurrence of f in this formula traces directly back to the source map; changing f while holding β fixed produces a different pullback tensor on V.

Pullback of a General Covariant k-Tensor

(f^{*} T) (v_{1}, \dots, v_{k}) = T (f (v_{1}), \dots, f (v_{k}))

The pattern extends to any covariant k-tensor T on W: the source map f is applied to every one of the k arguments before T is evaluated. The source map therefore appears once per tensor slot, and its role is uniform across tensors of any covariant rank.

Diagram of the Source Map and Its Reversed Pullback

Consequences of Properties of the Source Map

Injective Source Maps Preserve Nondegeneracy Information

If the source map f is injective, the pullback of a nondegenerate bilinear form on W need not itself be nondegenerate on V unless f is also surjective; but injectivity of f does guarantee that no nonzero vector of V is sent to zero, which is the first necessary condition for the pullback to have any chance of retaining nondegeneracy.

Composability Along Composed Source Maps

(g \circ f)^{*} = f^{*} \circ g^{*}

If two source maps f : V → W and g : W → U are composed to give g ∘ f : V → U, the pullback along the composite equals the pullback along f applied after the pullback along g, in reversed order. This contravariant composition law is a direct structural consequence of how the source map's direction determines the pullback's reversed direction, and it holds for any choice of source maps satisfying the composability condition g ∘ f.

Identity Source Map Yields the Identity Pullback

When the source map is the identity map id_V : V → V, the pullback operation (id_V)* acts as the identity on every covariant tensor, since applying id_V to each argument before evaluation changes nothing. This serves as the base case anchoring the general composition law and confirms the pullback construction behaves consistently with the most trivial possible choice of source map.

Distinguishing the Source Map from the Tensor Being Pulled Back

Two Independent Pieces of Data

A pullback operation always involves two separate inputs: the source map f, fixed once and used repeatedly, and the specific tensor T on W being transported, which may vary from one application to the next while f remains the same. Confusing these two roles — for instance, describing "the pullback" without specifying which source map was used — leaves the construction incompletely specified, since the same tensor T produces different pullbacks under different choices of source map.

The Source Map Is Never Itself Pulled Back

Only covariant tensors on W are pulled back via f; the source map f itself is not an object living in the target of its own pullback operation, but rather the fixed mechanism generating that operation. This distinguishes the role of the source map from the role of the tensors it acts upon throughout the entire theory of tensor pullback.