6.9 Tensor Contravariant Order Classification

Tensor Contravariant Order Classification is the scheme by which tensors are grouped according to the count of upper (contravariant) indices carried by their components, this count being called the contravariant order or contravariant rank of the tensor. A tensor with contravariant order p accepts exactly p one-form 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 p factors of the direct Jacobian matrix, one for every upper index. The contravariant order is independent of the covariant order and together with it fixes the tensor's full type, but classification by contravariant order alone groups tensors by how many one-form slots they consume, regardless of how many vector slots they also consume.

The Meaning of Contravariant Order

Contravariant Order as a Count of Upper Indices

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

Distinguishing Contravariant Order From Total Type

The full type of a tensor is the pair consisting of contravariant order and covariant order together, whereas the contravariant order classification isolates only the first member of that pair. Two tensors can share the same contravariant order while differing in covariant order, and such tensors are placed in the same contravariant-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 one-forms, such as symmetry properties restricted to the upper indices, independently of how it consumes vectors.

Classification by Increasing Contravariant Order

Contravariant Order Zero

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

Contravariant Order One

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

Contravariant Order Two

A tensor of contravariant order two carries two upper indices and consumes two one-form arguments, returning a scalar that depends linearly on each. The inverse metric tensor is the central example: it accepts two one-forms and returns a scalar built from their components, and it transforms with two factors of the direct Jacobian, one for each upper index.

Contravariant order two tensors need not be symmetric like the inverse metric; antisymmetric contravariant order two tensors, such as certain bivectors built by wedging two vectors, belong to the same class while carrying different internal symmetry.

Contravariant Order Greater Than Two

Tensors of contravariant order three or higher, such as certain multivectors accepting several one-forms, follow the identical pattern: one direct-Jacobian factor per upper index, contracted against the corresponding number of dummy indices in the original components. There is no upper bound on contravariant order in principle, since any positive integer number of one-form 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 contravariant order, provided they also share the same covariant order, can be added together and scaled by numbers to produce another tensor of the identical contravariant order. Contravariant order is therefore a grading that is preserved by the vector space operations native to the space of tensors of fixed type, making the contravariant-order classes into a natural stratification of that space.

Behavior Under Tensor Product

Forming the tensor product of two tensors adds their contravariant orders together, since every upper index from each factor is inherited independently by the product. A contravariant order two tensor combined by tensor product with a contravariant order one tensor yields a contravariant 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 contravariant order by one for every upper index removed in this way, exactly as it reduces the covariant order by one for the paired lower index removed. Lowering an upper index with the metric decreases the contravariant order by one and simultaneously increases the covariant order by one, moving the tensor from one contravariant-order class into the adjacent lower class.

Role in Symmetric and Antisymmetric Subclassification

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