1.2.46 Covariant Order Definition

Covariant Order Definition is the characterization of the number of covariant, or lower, index slots possessed by a tensor, counted independently of any contravariant slots the same tensor may also carry. For a tensor of type $(r,s)$, the covariant order is the second entry $s$, and it records specifically how many copies of the dual space contribute to the tensor, independently of how many copies of the vector space itself are also present.

Formal Definition

Let $V$ be a vector space over a field $F$, with dual space $V^{*}$. For a tensor

the covariant order of $T$ is the integer $s$, the number of factors of $V^{*}$ appearing in the tensor product space in which $T$ lives. In component notation, the covariant order equals the number of subscript indices carried by the tensor,

since, by the notational convention pairing subscripts with covariant slots, exactly $s$ indices appear below the line regardless of how many appear above it.

Covariant Order as an Independent Count

The defining feature of covariant order is that it is tracked separately from contravariant order and does not depend on it. A tensor of type $(3,1)$ and a tensor of type $(0,1)$ have the same covariant order, one, despite differing substantially in total order and in contravariant order; both accept exactly one vector argument when viewed as multilinear maps, even though only the second is a covector by itself in the narrowest sense. This independence is what makes covariant order a useful invariant on its own, distinct from total order: it isolates precisely how many vector arguments — as opposed to covector arguments — a tensor requires.

Covariant Order in Special Cases

Zero Covariant Order

A tensor with covariant order zero is purely contravariant, of type $(r,0)$, and accepts no vector arguments at all when viewed as a multilinear map — only covector arguments, or none, if $r$ is also zero.

Covariant Order One

A tensor with covariant order one and contravariant order zero is a linear functional, an element of $V^{*}$. More generally, any tensor with covariant order one, regardless of its contravariant order, accepts exactly one vector argument among its inputs.

Purely Covariant Tensors

When contravariant order is also zero, a tensor's covariant order coincides with its total order, and the tensor is simply a multilinear form of that order on $V$; a covariant order of two, in this purely covariant case, describes a bilinear form, and a covariant order of $s$ in general describes an $s$-linear form.

Behavior Under Basic Operations

Tensor Product

Forming the tensor product of two tensors adds their covariant orders together, independently of what happens to contravariant order: a tensor of covariant order $s_1$ combined with a tensor of covariant order $s_2$ produces a tensor of covariant order $s_1+s_2$.

Contraction

Contracting a matched contravariant and covariant index pair reduces covariant order by exactly one, together with a corresponding reduction of contravariant order by one, since contraction always removes one slot of each kind simultaneously; there is no operation within ordinary tensor algebra that reduces covariant order alone while leaving contravariant order untouched, since a valid contraction requires pairing a lower index with an upper one.

Role Within Tensor Algebra

Covariant order provides the half of a tensor's type that specifically tracks its dependence on vector arguments, complementing contravariant order, which tracks dependence on covector arguments. Isolating covariant order is useful whenever attention is restricted to how a tensor behaves as a function of vectors alone — as when studying purely covariant subalgebras such as the algebra of multilinear forms, symmetric tensors, or differential forms, all of which are most naturally graded by covariant order rather than by the finer type or the coarser total order.