11.15.2 Tensor Dual Transformation Pullback Relation

The Tensor Dual Transformation Pullback Relation defines how dual tensors transform under coordinate changes via linear mappings in algebraic structures.

Tensor Dual Transformation Pullback Relation is the identity connecting the pullback operation on covectors, which moves a covector backward through a coordinate map, to the covariant transformation law, showing that the transformation of covariant tensor components under a change of coordinates is exactly an instance of pulling the covector back along the coordinate transformation.

Definition and Setting

The Pullback Operation Described

Given a smooth map between coordinate systems, the pullback operation takes a covector defined using the new coordinates and produces a corresponding covector expressed in terms of the old coordinates, by composing the covector with the derivative of the coordinate map.

(φ^{*} ω) (v) = ω (D φ (v))

The Relation to Covariant Transformation

Applying the pullback to the dual basis covectors of the new coordinate system reproduces exactly the covariant transformation law for lower index components, showing that lowering an index under a coordinate change and pulling back a covector along that same coordinate change are the same operation viewed from two different notational traditions.

A_{i^{'}} = {(φ^{*} A)}_{i^{'}} = \frac{\partial x^{j}}{\partial x^{i^{'}}} A_{j}

Why Covectors Pull Back While Vectors Push Forward

The Direction of the Pullback Matches the Covariant Direction

The pullback moves covectors in the direction opposite to the coordinate map itself, taking objects defined in the new coordinates back to the old coordinates, and this backward direction of travel matches precisely the inverse Jacobian factor that characterizes covariant transformation.

Contrast With the Pushforward of Vectors

Vectors, by contrast, are moved forward along the coordinate map through the pushforward operation, using the derivative of the map applied directly rather than through composition on the covector side, and this forward direction matches the direct Jacobian factor characterizing contravariant transformation.

Consequences of the Pullback Relation

Explaining Covariant Transformation Through a Geometric Operation

Recognizing covariant transformation as a pullback provides a geometric explanation for the inverse Jacobian factor, framing it not as an arbitrary algebraic rule but as the natural consequence of moving a linear functional backward through a coordinate map via composition with the map's derivative.

Compatibility With Composition of Coordinate Changes

The pullback relation is compatible with composing several coordinate changes in sequence, since pulling a covector back along a composed map is the same as pulling it back along each individual map in the reverse order, matching the corresponding composition rule already established for the covariant transformation law.

Role Within Tensor Algebras

Bridging Tensor Notation and Differential Geometric Language

The pullback relation connects the indexed component notation used throughout tensor algebra with the coordinate-free language used in differential geometry, showing that these are two descriptions of a single underlying operation on covectors under a change of coordinates.

Deepening the Understanding of Dual Transformation Behavior

By identifying covariant transformation with pullback and, implicitly, contravariant transformation with pushforward, the pullback relation deepens the understanding of dual transformation behavior, grounding the mirrored transformation rules in the geometric operations of pulling back and pushing forward along a coordinate map.