6.16 Tensor Type Zero Two Classification

Tensor Type Zero Two Classification is the categorization of a tensor as a purely covariant object built from two copies of the dual space, meaning it possesses two lower indices and no upper indices, and it is constructed as an element of V* ⊗ V* rather than involving the original vector space V in any of its slots. This classification places the tensor in the family of objects that take two vectors as input and return a scalar, functioning as a bilinear form directly on V, and it is the classification underlying some of the most important structures in tensor algebra, most notably the metric tensor that defines lengths and angles.

Defining Features of the Type Zero Two Class

Index Structure

A type (0,2) tensor T is written with two subscripts, T_{ij}, indicating that both indices are covariant. Coordinate-free, T is an element of:

with no factor of V appearing anywhere in the product, in direct contrast to the type (2,0) classification, which uses two factors of V, and to the type (1,1) classification, which mixes one factor of each.

Action as a Bilinear Map on Vectors

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

extended by linearity to general elements of V* ⊗ V*. This role, taking two vectors and producing a scalar, is exactly the pattern used to define inner products, and it is why type (0,2) tensors are the natural algebraic home of metrics.

Transformation Law Characterizing the Classification

Both Indices Transform Covariantly

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 (0,2) tensor transform using the forward matrix A on both indices:

This double application of A, with no factor of B, is the defining transformation signature of the classification, opposite in character to the type (2,0) case, which uses B on both indices.

Consequence for Congruence Rather than Similarity

In matrix language, this transformation law is congruence, T' = A^T T A, rather than the similarity transformation T' = A^{-1} T A that governs type (1,1) operators. This distinction matters algebraically: congruent matrices can have different eigenvalues, since congruence is not the same equivalence relation as similarity, but they always share the same signature, meaning the number of positive, negative, and zero eigenvalues is a congruence invariant even though the eigenvalues themselves are not.

Canonical Examples Within This Classification

The Metric Tensor

The most important type (0,2) tensor is the metric g_{ij}, which defines an inner product on V by g(v, w) = g_{ij} v^i w^j, and which is required to be symmetric, g_{ij} = g_{ji}, and typically nondegenerate, meaning its associated matrix is invertible. The metric is the tensor that makes possible the raising and lowering of indices, converting between the type (2,0), (1,1), and (0,2) classifications.

The Second Fundamental Form and Other Bilinear Forms

Other classical examples in this classification include the second fundamental form of a surface, which measures curvature, and any general symmetric or antisymmetric bilinear form used to define quadratic expressions or oriented area elements on V itself, mirroring the constructions available for type (2,0) tensors but applied to vectors rather than covectors.

Diagram of the Two Lower Index Structure

Distinguishing This Classification from Related Types

Versus Type Two Zero

A type (2,0) tensor acts on covectors and transforms with two factors of B, whereas the type (0,2) classification acts on vectors and transforms with two factors of A. The metric provides the canonical bridge between the two: given a nondegenerate g_{ij}, its inverse g^{ij} is a type (2,0) tensor, and the two are related by g^{ik} g_{kj} = δ^i_j.

Versus Type One One

A type (1,1) tensor mixes A and B, giving it the role of a linear operator, whereas the pure covariance of type (0,2) excludes it from mapping V to V directly; instead, a type (0,2) tensor can be converted into a type (1,1) operator only with the help of an auxiliary type (2,0) tensor, typically the inverse metric, to raise one of its two indices.

Rank and Order Terminology

The notation (0,2) records the tensor's order, two, entirely covariant. This order is independent of the dimension n of V, though the number of independent components, n² before any symmetry is imposed, does depend on it, matching exactly the component count of the type (2,0) classification despite the two classifications playing structurally opposite roles.