6.17.1 Tensor p q Covariant Count

Tensor p q Covariant Count is the value q appearing in the type notation (p, q), recording exactly how many lower, covariant indices a tensor possesses, and thereby specifying how many independent covector-shaped slots the tensor provides for accepting vectors as input. This count is the complementary half of the complete type specification alongside the contravariant count p, and it determines how many vector arguments a tensor, viewed as a multilinear map, is able to consume directly.

What the Covariant Count Measures

Number of Lower Index Slots

For a tensor T^{i_1 ... i_p}_{j_1 ... j_q}, the covariant count q is the number of subscript positions, j_1 through j_q. Each such slot, when the tensor acts as a multilinear map, is filled by a vector from V, so a type (p, q) tensor takes q vectors and p covectors as its total input and returns a scalar, or, if some indices remain free, a tensor of reduced type.

Contribution to Total Component Count

If the underlying vector space has dimension n, the covariant count contributes a factor of n^q to the total component count n^{p+q}, since each of the q lower indices ranges independently over the n basis directions.

Transformation Behavior Tied to the Covariant Count

Each Lower Index Uses the Forward Transition Matrix

Under a change of basis with transition matrix A, every one of the q lower indices picks up its own independent factor of A:

This is the origin of the term "covariant": each associated index transforms in the same direction, using the same matrix A, as the basis vectors themselves, which is the opposite behavior from the contravariant indices governed by B.

Consistency Check Using Covector Components

The simplest instance of this rule, q = 1 and p = 0, reduces exactly to the transformation law for the components of an ordinary covector, φ'_j = A^l_j φ_l, confirming that the general covariant count formula extends the familiar covector transformation law to any number of lower indices.

The Covariant Count Under Algebraic Operations

Tensor Product Adds Counts

If S has covariant count q and T has covariant count s, their tensor product S ⊗ T has covariant count q + s, since the lower indices of S and the lower indices of T are simply concatenated without interaction.

Contraction Removes One Lower Index

Contracting one of the q lower indices of a type (p, q) tensor against one of its own upper indices, when p ≥ 1, produces a tensor with covariant count q - 1. This is why the trace of a type (1,1) tensor, which has covariant count 1, results in a scalar with covariant count 0.

Lowering an Index Increases the Count

Given a metric g_{ij}, contracting it against one upper index of a tensor lowers that index, converting a contravariant slot into a covariant one and increasing q by one while decreasing p by one, again leaving the total order p + q unchanged.

Diagram of the Covariant Slots

Distinguishing the Covariant Count from the Contravariant Count

Independent Yet Complementary Roles

The covariant count q is set independently of p when defining a tensor's type, and knowing q alone does not fully classify a tensor: a covariant count of q = 2 is shared by a purely covariant type (0,2) bilinear form and by a type (1,2) tensor that also carries an upper index, and these two objects play very different roles despite matching in covariant count.

Covariant Count of Zero

When q = 0, the tensor has no lower indices at all, and it is purely contravariant, meaning it cannot directly consume any vector as an argument without the assistance of an auxiliary covariant structure such as a metric to lower one of its indices first; this is the case for the inverse metric g^{ij} and for ordinary vectors themselves.