6.15 Tensor Type Two Zero Classification

Tensor Type Two Zero Classification is the categorization of a tensor as a purely contravariant object built from two copies of a vector space, meaning it possesses two upper indices and no lower indices, and it is constructed as an element of the tensor product V ⊗ V rather than involving the dual space V* in any of its slots. This classification places the tensor in the family of objects that take two covectors as input and return a scalar, or equivalently that can be viewed as a bilinear map on the dual space, distinguishing it sharply from mixed tensors and from purely covariant tensors in how its components behave under a change of basis.

Defining Features of the Type Two Zero Class

Index Structure

A type (2,0) tensor T is written with two superscripts, T^{ij}, indicating that both indices are contravariant. In the coordinate-free formulation, T is an element of:

with no factor of V* appearing anywhere in the product. This is what separates the type (2,0) classification from the type (1,1) classification, which mixes one factor of V with one factor of V*, and from the type (0,2) classification, which uses two factors of V*.

Action as a Bilinear Map on Covectors

Every type (2,0) tensor defines a bilinear map on pairs of covectors: given φ and ψ in V*, the tensor T = v ⊗ w acts as:

extended by linearity to general elements of V ⊗ V. This shows the classification is not merely a labeling convention but corresponds to a concrete functional role: type (2,0) tensors are precisely the bilinear forms defined on the dual space.

Transformation Law Characterizing the Classification

Both Indices Transform Contravariantly

Under a change of basis with transition matrix A, where new basis vectors satisfy e'_i = A^k_i e_k, the components of a type (2,0) tensor transform using the inverse matrix B = A^{-1} on both indices:

This double application of B is the defining signature of the classification: no index uses the forward matrix A, which is what makes the tensor purely contravariant. This is qualitatively different from the type (1,1) case, in which one index uses A and the other uses B, and from the type (0,2) case, in which both indices use A.

Consequence for Symmetry Properties

Because both indices transform identically, the symmetric and antisymmetric parts of a type (2,0) tensor, defined by:

remain symmetric or antisymmetric in every basis, since swapping i and j commutes cleanly with the transformation law when both indices are subject to the same kind of matrix factor. This basis-independence of symmetry is a defining structural consequence of belonging to the type (2,0) classification.

Canonical Examples Within This Classification

Products of Vectors

The simplest type (2,0) tensors are the elementary products v ⊗ w of two vectors, whose components are T^{ij} = v^i w^j. Sums of such elementary products span the entire space V ⊗ V, and every type (2,0) tensor can be written, though not uniquely, as a finite sum of such products.

The Inverse Metric

When a vector space is equipped with a metric tensor g_{ij}, which is a type (0,2) object, its matrix inverse g^{ij} is a canonical type (2,0) tensor, satisfying g^{ik} g_{kj} = δ^i_j. The inverse metric is used to raise indices, converting covariant components into contravariant components, and it is the standard example demonstrating that type (2,0) tensors arise naturally even in contexts dominated by covariant objects.

Diagram of the Two Upper Index Structure

Distinguishing This Classification from Related Types

Versus Type Zero Two

A type (0,2) tensor, such as g_{ij}, consumes two vectors and returns a scalar, and its components transform with two factors of A rather than B. Type (2,0) and type (0,2) tensors are related by the metric, which converts one into the other, but as raw classifications they behave oppositely under change of basis and serve dual purposes: (2,0) tensors act on covectors, (0,2) tensors act on vectors.

Versus Type One One

A type (1,1) tensor uses one factor of A and one factor of B, giving it the role of a linear operator rather than a bilinear form. The complete absence of any lower index in the type (2,0) classification is precisely what excludes it from acting as a map from V to V, confining its natural domain to pairs of covectors instead.

Rank and Order Terminology

The numbers in the notation (2,0) denote the tensor's order, two, split as two contravariant slots and zero covariant slots. This order should not be confused with the dimension of the underlying vector space, since the type (2,0) classification applies uniformly regardless of whether V is two-dimensional, three-dimensional, or of any other finite dimension, with only the number of independent components, equal to the dimension squared, depending on the dimension of V.