4.7.4 Tensor Mixed Vector Space Domain

Tensor Mixed Vector Space Domain is the case of a tensor's multilinear domain structure in which the factor spaces are drawn from more than one distinct variance class simultaneously, most commonly a combination of the base space $V$ and its dual $V^{*}$, so that some argument slots accept vectors while others accept covectors within the same domain. It is the structural condition underlying every tensor of type $(r, s)$ with both $r > 0$ and $s > 0$, and it is what forces a tensor's transformation law to combine two opposite behaviors, one from each variance class, rather than a single uniform one.

Formal Definition

Combining Two Variance Classes

A mixed vector space domain for a type $(r, s)$ tensor with $r, s \geq 1$ takes the form

Both blocks repeat their respective space internally (making each block a repeated vector space domain in its own right), but the domain as a whole mixes the two distinct spaces $V$ and $V^{*}$ together, distinguishing it from a domain built from a single repeated space alone.

Type (1,1) as the Minimal Mixed Domain

The smallest genuinely mixed domain occurs at type $(1,1)$: a single contravariant slot bound to $V^{}$ paired with a single covariant slot bound to $V$, giving the domain $V^{} \times V$. A multilinear map on this domain is equivalent, via the canonical pairing, to a linear endomorphism of $V$, illustrating how mixing variance classes even minimally already produces objects with a different character than a purely covariant or purely contravariant tensor.

Structural Consequences of Mixing

Two Independent Symmetrization Blocks

Because permutation of arguments is only well-typed among slots sharing the same factor space, a mixed domain supports symmetrization or antisymmetrization independently within its contravariant block of $r$ slots and independently within its covariant block of $s$ slots, but not across the two blocks without additional structure. This is the direct consequence of the domain's mixed composition: the two blocks behave as separate repeated vector space domains glued together.

Opposite Transformation Behavior

Under a change of basis given by a matrix $A$, the $s$ covariant slots transform using $A$ while the $r$ contravariant slots transform using $A^{-1}$ (or its transpose, by convention). A mixed domain is precisely what forces both transformation rules to appear together in the same tensor's transformation law, in contrast to a purely covariant or purely contravariant domain, where only one rule applies throughout.

Contraction Between the Two Blocks

The canonical pairing between $V$ and $V^{*}$ makes contraction possible precisely between a contravariant slot from the $r$-block and a covariant slot from the $s$-block, reducing arity by two and producing a new mixed (or possibly pure) domain with $r - 1$ contravariant and $s - 1$ covariant slots. This cross-block contraction is unavailable within a purely repeated domain of a single variance, since there is no canonical pairing of $V$ with itself without an externally supplied metric.

Applications of Mixed Domains

Linear Endomorphisms as Type (1,1) Tensors

Every linear map $L : V \to V$ corresponds to a unique type $(1,1)$ tensor on the mixed domain $V^{*} \times V$, via $T(\phi, v) = \phi(L(v))$; this identification is the reason type $(1,1)$ tensors are often described interchangeably as either bilinear pairings or as linear operators.

Curvature and Torsion Tensors

Geometric tensors such as the Riemann curvature tensor, typically presented in type $(1,3)$ form, live on a mixed domain combining one contravariant slot with three covariant slots; the mixed structure is essential to how curvature simultaneously encodes a linear operator on tangent vectors (via the single upper index) and a genuinely multilinear dependence on three directional inputs (via the three lower indices).

Trace as a Mixed-Domain Contraction

The trace of a linear operator, viewed as the type $(1,1)$ tensor $T(\phi, v) = \phi(Lv)$, is exactly the contraction of its single contravariant slot against its single covariant slot; this is the most elementary example of how a mixed vector space domain enables an operation, contraction between blocks, that is unavailable in a domain built from only one variance class.

Summary of Key Points

• A mixed vector space domain combines slots bound to $V$ and slots bound to $V^{*}$ within a single tensor's domain structure.
• The smallest genuinely mixed domain, type $(1,1)$, identifies multilinear maps on $V^{*} \times V$ with linear endomorphisms of $V$.
• Symmetrization applies independently within the contravariant block and within the covariant block, but not across the two.
• Mixed domains force opposite transformation behaviors, $A$ and $A^{-1}$, to appear together in the same tensor's change-of-basis law.
• Contraction between the two blocks, made possible by the canonical pairing of $V$ and $V^{*}$, generalizes the trace of a linear operator.