12.2.5 Tensor Mapping Operation Area

Tensor Mapping Operation Area is the subset of tensor algebra concerned with transporting tensors between two distinct vector spaces connected by a linear map, through the paired procedures of pushforward and pullback, addressing situations that fall outside the scope of the other operation areas, which act within a single, fixed vector space rather than relating two separate ones.

Foundational Setting

A Different Kind of Relationship Between Tensors

The additive, composition, and evaluation areas of tensor algebra all concern tensors defined relative to one fixed vector space. The mapping area instead addresses what happens when a linear map $$f:V\to W$$ connects two spaces that may differ in dimension or structure, and a tensor defined on one of them needs to be related to an object on the other.

Two Natural Directions of Transport

Because a linear map has a natural forward direction and only sometimes an inverse, two distinct transport procedures arise: pushforward, carrying contravariant objects from $$V$$ to $$W$$, and pullback, carrying covariant objects from $$W$$ back to $$V$$.

Pushforward

Transporting Contravariant Objects Forward

Pushforward applies the map directly to a vector's components, carrying it from the domain space into the codomain space:

No Invertibility Needed for This Direction

Because pushforward operates in the same direction as the map itself, it remains well defined for any linear map, whether or not that map is invertible or even dimension-preserving between the two spaces.

Pullback

Transporting Covariant Objects Backward

Pullback carries a covector defined on the codomain space back to the domain space by precomposing it with the map:

Also Requiring No Invertibility

Like pushforward, pullback for a purely covariant object requires only the map's own components, applying naturally in the reverse direction regardless of whether the original map admits an inverse.

Visual Overview

Diagram of the Two Mapping Directions

Mixed Tensors and the Need for Invertibility

Combining Both Directions

A mixed tensor, carrying both upper and lower indices, requires pushforward applied to its contravariant indices and pullback applied to its covariant indices simultaneously if it is to be transported as a single, complete object rather than in separate pieces.

Why the Map Must Be Invertible Here

Consistently transporting such a mixed tensor in one fixed direction, say entirely from $$W$$ back to $$V$$, requires applying the inverse map to its contravariant indices, which is only possible when $$f$$ is invertible, marking the point at which the mapping area's two basic procedures must be supplemented by this additional structural requirement.

Relationship to Ordinary Basis Change

A Special Case Recovers Familiar Behavior

When the two spaces $$V$$ and $$W$$ coincide and the map $$f$$ is simply an invertible change of basis on that single space, pushforward and pullback reduce exactly to the familiar contravariant and covariant transformation laws studied elsewhere in tensor algebra, showing that ordinary basis change is a particular instance of the more general mapping area rather than an entirely separate phenomenon.

Summary of Key Traits

Defining Characteristics

• The mapping area transports tensors between two distinct vector spaces connected by an explicit linear map, rather than acting within a single fixed space.
• Pushforward carries contravariant objects forward along the map, and pullback carries covariant objects backward, neither requiring the map to be invertible.
• Transporting a mixed tensor as a single combined object requires the connecting map to be invertible.
• Ordinary basis change within one space is recovered as the special case where the two spaces coincide and the map is an invertible self-map.