6.8 Tensor Covariant Order Classification

Tensor Covariant Order Classification is the scheme by which tensors are grouped according to the count of lower (covariant) indices carried by their components, this count being called the covariant order or covariant rank of the tensor. A tensor with covariant order q accepts exactly q vector arguments and returns a scalar through a map that is linear in each argument separately, and its components acquire, under a change of coordinates, exactly q factors of the inverse Jacobian matrix, one for every lower index. The covariant order is independent of the contravariant order and together with it fixes the tensor's full type, but classification by covariant order alone groups tensors by how many vector slots they consume, regardless of how many one-form slots they also consume.

The Meaning of Covariant Order

Covariant Order as a Count of Lower Indices

Every lower index attached to a tensor's components signals one argument slot that must be filled with a vector before a number is produced, and signals one factor of the inverse Jacobian in the transformation law. Counting these lower indices, independently of any upper indices present, gives the covariant order. A tensor written with components carrying three subscripts and one superscript, for instance, has covariant order three and contravariant order one, and belongs to the covariant-order-three class regardless of the fact that it also carries a single upper index.

Distinguishing Covariant Order From Total Type

The full type of a tensor is the pair consisting of contravariant order and covariant order together, whereas the covariant order classification isolates only the second member of that pair. Two tensors can share the same covariant order while differing in contravariant order, and such tensors are placed in the same covariant-order class even though their overall types differ. This coarser grouping is useful whenever the property under study depends only on how the tensor consumes vectors, such as symmetry properties restricted to the lower indices, independently of how it consumes one-forms.

Classification by Increasing Covariant Order

Covariant Order Zero

A tensor of covariant order zero carries no lower indices at all and therefore consumes no vector arguments. If its contravariant order is also zero it is a scalar; if it carries only upper indices it is a purely contravariant tensor, such as a vector or a bivector built from vectors. Covariant order zero marks the absence of any dependence on vector inputs.

Covariant Order One

A tensor of covariant order one carries a single lower index and consumes exactly one vector argument. The prototypical example is the one-form, or covector, which assigns a scalar to each vector through a single linear pairing. Its transformation law involves exactly one factor of the inverse Jacobian.

Covariant Order Two

A tensor of covariant order two carries two lower indices and consumes two vector arguments, returning a scalar that depends linearly on each. The metric tensor is the central example: it accepts two vectors and returns their inner product, and it transforms with two factors of the inverse Jacobian, one for each lower index.

Covariant order two tensors need not be symmetric like the metric; antisymmetric covariant order two tensors, such as electromagnetic field tensors expressed with lowered indices, belong to the same class while carrying different internal symmetry.

Covariant Order Greater Than Two

Tensors of covariant order three or higher, such as certain curvature-derived objects with all indices lowered, or general multilinear forms accepting several vectors, follow the identical pattern: one inverse-Jacobian factor per lower index, contracted against the corresponding number of dummy indices in the original components. There is no upper bound on covariant order in principle, since any positive integer number of vector slots can in principle be accommodated by a sufficiently high-order multilinear form.

Structural Consequences of the Classification

Closure Under Addition and Scalar Multiplication

Tensors sharing the same covariant order, provided they also share the same contravariant order, can be added together and scaled by numbers to produce another tensor of the identical covariant order. Covariant order is therefore a grading that is preserved by the vector space operations native to the space of tensors of fixed type, making the covariant-order classes into a natural stratification of that space.

Behavior Under Tensor Product

Forming the tensor product of two tensors adds their covariant orders together, since every lower index from each factor is inherited independently by the product. A covariant order two tensor combined by tensor product with a covariant order one tensor yields a covariant order three tensor, with no interaction or cancellation between the index counts of the two factors.

Behavior Under Contraction

Contracting an upper index of a tensor against a lower index of the same or another tensor reduces the covariant order by one for every lower index removed in this way, exactly as it reduces the contravariant order by one for the paired upper index removed. Raising a lower index with the inverse metric decreases the covariant order by one and simultaneously increases the contravariant order by one, moving the tensor from one covariant-order class into the adjacent lower class.

Role in Symmetric and Antisymmetric Subclassification

Within a fixed covariant order class of two or more, tensors further subdivide according to how their lower indices behave under permutation: totally symmetric covariant tensors, totally antisymmetric covariant tensors, which are the differential forms of that order, and tensors with mixed symmetry that fit neither extreme. This finer subclassification is only meaningful once the covariant order has already been fixed, since permutation symmetry is a relation among indices of the same variance type acting on the same footing.