3.16 Tensor Covector Pullback Operation

Tensor Covector Pullback Operation is the operation by which a covector defined on the codomain of a linear map is converted into a covector on the domain, by composing the covector with the linear map itself. Given a linear map f: V → W and a covector φ ∈ W*, the pullback of φ along f, written f*φ, is the covector on V defined by (f*φ)(v) = φ(f(v)) for all v ∈ V. This operation is the fundamental building block of the dual map construction and extends naturally to pulling back covariant tensors of any order, not just single covectors.

Definition of the Pullback

Pullback of a Single Covector

Given f: V → W and φ ∈ W*, the pullback f*φ ∈ V* is defined by

The pullback is well-defined for any linear map f, regardless of whether f is injective, surjective, or neither, because it only requires composing φ with f, an operation that is always possible for any linear map and any covector on its codomain.

Linearity of the Pullback Map

For fixed f, the assignment φ ↦ f*φ is itself linear in φ, satisfying f*(aφ + bψ) = a(f*φ) + b(f*ψ) for scalars a, b and covectors φ, ψ ∈ W*. This linearity is what makes the pullback a linear map f*: W* → V* in its own right, the dual map associated to f.

Coordinate Description

Component Formula

If f has matrix A = (A^i_j) relative to bases e_1, ..., e_n of V and ᵤ_1, ..., ᵤ_m of W, and φ has components φ_i relative to the dual basis of W*, then the components of the pullback f*φ relative to the dual basis of V* are given by

using the Einstein summation convention with the index i summed from 1 to m. This formula makes explicit that pulling back a covector is a matrix-vector multiplication using the transpose of the matrix of f, matching the general fact that dual maps are represented by transpose matrices.

Example in Low Dimension

If V and W are both two-dimensional, f is given by a 2 × 2 matrix A, and φ has components (φ_1, φ_2), the pullback f*φ has components obtained by multiplying the row vector (φ_1, φ_2) by the matrix A, producing a new row vector representing f*φ in the basis dual to that of V.

Extension to Covariant Tensors

Pullback of a Bilinear Form

The pullback operation extends beyond single covectors to covariant tensors of any order. If g is a type (0, 2) tensor on W, meaning a bilinear form taking two vectors of W and returning a scalar, its pullback f*g is the bilinear form on V defined by

for all v_1, v_2 ∈ V. The same idea generalizes to a type (0, q) tensor T on W: the pullback f*T is the type (0, q) tensor on V given by evaluating T on the images under f of q vectors from V.

Restriction to Purely Covariant Tensors

The pullback operation, in this general form, applies only to purely covariant tensors, those of type (0, q), because it relies on composing arguments with f before feeding them into T, which requires T to accept vectors, not covectors, as its inputs. Tensors with contravariant slots require the complementary pushforward operation instead, which is only canonically defined when f is invertible.

Functorial Behavior of the Pullback

Reversal Under Composition

The pullback operation obeys the same order-reversal law as the general dual map: for f: V → W and g: W → U, and a covariant tensor T on U,

Pulling back along a composite map is the same as pulling back first along g and then along f, in that order, mirroring the contravariant functoriality established for the general dual map.

Pullback of the Identity

Pulling back a covariant tensor along the identity map leaves it unchanged: (id_V)*T = T for any covariant tensor T on V, since composing with the identity map does not alter the arguments fed into T.

Diagrammatic Summary

The diagram shows the map f carrying V into W, the covector φ acting on W, and the pulled-back covector f*φ acting on V, produced by routing every input through f before applying φ.