3.18.3 Tensor Covariant Index Placement

Tensor Covariant Index Placement is the convention governing where, as subscripts rather than superscripts, indices belonging to covariant tensor slots are written, together with the transformation rule that placement is meant to encode: that covariant components change under a basis transformation in the same direction as the basis vectors themselves. For a covector ω on a vector space V with basis {eⱼ}, its components ωᵢ are placed with a lower index, and this single typographic choice fixes, for every subsequent formula, how ωᵢ must transform, how it may be legally contracted, and what geometric or algebraic role it plays relative to contravariant, upper-indexed quantities.

The Defining Transformation Rule

Same-Direction Transformation as the Basis

If the basis of V is changed by an invertible matrix A, so that ẽⱼ = Aᵏⱼeₖ, covariant components transform by the same matrix A, not its inverse:

This is precisely why the term covariant is used: the components co-vary, meaning they transform in step with the basis vectors, in contrast to contravariant vector components vʲ, which transform by A⁻¹, against the direction of the basis change.

Why Lower Placement Encodes This Rule

The choice to write covariant indices as subscripts is not arbitrary decoration; it is a systematic labeling scheme so that, upon seeing an index in the lower position throughout a derivation, one can immediately invoke the same-direction transformation law without re-deriving it. This labeling discipline is what allows the Einstein summation convention to double as a correctness check: a sum is basis-independent precisely when it contracts one lower, covariant index against one upper, contravariant index.

Covariant Index Placement in the Dual Basis

Basis Covectors Carry Upper Indices, Not Lower

A subtlety of covariant index placement concerns the dual basis vectors {eⁱ} themselves, which carry upper, not lower, indices, even though they are elements of the dual space V* whose general elements ω are indexed with lower indices. This apparent inversion is consistent: the dual basis covectors eⁱ transform contragrediently to the components ωᵢ precisely so that the full expansion ω = ωᵢeⁱ, with one lower and one upper repeated index, is itself basis-independent as a whole object, even though its two factors individually depend on the basis.

Consistency Check via Biorthogonality

The relation eⁱ(eⱼ) = δⁱⱼ fixes the placement scheme: the free index on eⁱ is upper, the free index on eⱼ is lower, and the result δⁱⱼ carries one of each, matching the general rule that a well-formed tensorial equation has identical free-index placement, upper or lower, on both sides.

Covariant Placement Under Pullback

Index Placement in the Pullback Formula

For a linear map T: V → W with components Tⁱⱼ, where i indexes the codomain W and j indexes the domain V, the pullback of a covector ω on W, with covariant components ωᵢ, is given by

The free index on the left, j, is lower, matching the lower placement on the right, confirming that T*ω is itself a genuinely covariant object on V, consistently indexed with a subscript exactly as ω was on W.

Why the Placement Must Be Preserved Across Pullback

If the pullback formula produced a result with an upper free index instead, this would signal that the construction failed to produce a covector at all, but rather some other type of tensor; covariant index placement therefore functions as a built-in type-check on the pullback operation, verifying at the notational level that the operation preserves the covariant character of the object being transported.

Distinguishing Covariant Placement from Mixed and Contravariant Placement

Contravariant Placement for Comparison

Vector components vʲ, and more generally contravariant tensor slots, are placed as superscripts, transforming by the inverse matrix A⁻¹ under the same change of basis that sends covariant indices via A. The deliberate visual contrast between subscript and superscript positions is the entire notational apparatus by which these two opposite transformation behaviors are kept distinguishable throughout lengthy calculations.

Mixed Tensors and Simultaneous Placement

A mixed tensor, such as the components Tⁱⱼ of a linear endomorphism regarded as a (1,1) tensor, carries both an upper, contravariant index and a lower, covariant index simultaneously, each transforming according to its own respective rule, A or A⁻¹, independently of the other. Covariant index placement remains meaningful even within such mixed objects, applying separately to each covariant slot regardless of how many contravariant slots the same tensor also possesses.

Practical Role of Placement in Tensor Contraction

Contraction Rules Determined by Placement

The rule that only a lower index may be validly contracted against a matching upper index of the same name is a direct consequence of covariant index placement combined with the corresponding contravariant convention; this rule guarantees that every legally written contraction, such as ωᵢvⁱ or ωᵢTⁱⱼ, automatically evaluates to a quantity whose remaining free indices, if any, transform correctly and whose value, if no free indices remain, is a true basis-independent scalar.

Placement in Higher-Rank Covariant Objects

For a tensor with several covariant slots, such as a bilinear form gᵢⱼ, every one of its indices is placed as a subscript, and each individually obeys the same-direction transformation law under a change of basis, so that the full object transforms as g̃ₖₗ = gᵢⱼAⁱₖAʲₗ, with the change-of-basis matrix A applied once for each covariant slot the tensor possesses.