1.2.53 Type Zero Two Tensor Definition

A Type Zero Two Tensor is a bilinear mapping from two vector spaces to a field, encoding linear relationships between pairs of vectors.

Type Zero Two Tensor Definition is the characterization of a tensor with no contravariant factors and exactly two covariant factors, the case in which the type $(r, s)$ equals $(0, 2)$ . A type $(0, 2)$ tensor is precisely a bilinear form on a vector space, and this type houses some of the most consequential tensors in mathematics and physics, including the metric tensor that supplies a vector space with a notion of length and angle.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . Setting $r = 0$ and $s = 2$ gives the tensor product space

T_{2}^{0} (V) = V^{*} \otimes V^{*}

An element $T$ of this space is a type $(0, 2)$ tensor, written in components as $T_{i j}$ with both indices as subscripts. Under the identification of tensor product spaces with spaces of multilinear maps, $T$ is equivalently a bilinear map

T : V \times V \to F

taking two vector arguments and returning a scalar, linear in each argument separately, matching exactly the general definition of a bilinear form.

Symmetric and Alternating Cases

Symmetric Type (0, 2) Tensors and the Metric

A symmetric type $(0, 2)$ tensor satisfies $T_{i j} = T_{j i}$ . When such a tensor is also nondegenerate, and in the geometric setting positive-definite, it is called a metric tensor and is denoted $g_{i j}$ , providing the algebraic structure by which lengths and angles are measured on $V$ ,

∥ v ∥ =^{2} = \sum_{i, j} g_{i j} v^{i} v^{j}

The metric tensor is the single most important example of a type $(0, 2)$ tensor and is the tensor used to lower a contravariant index into a covariant one throughout Riemannian and pseudo-Riemannian geometry.

Alternating Type (0, 2) Tensors

An alternating type $(0, 2)$ tensor satisfies $T_{i j} = - T_{j i}$ and is called a two-form, an element of the second exterior power of the dual space $⋀^{2} V^{*}$ . Two-forms measure signed area spanned by pairs of vectors and, when nondegenerate, define a symplectic structure on $V$ , in direct parallel with the way a symmetric, positive-definite type $(0, 2)$ tensor defines a metric structure.

Matrix Representation and Congruence

Relative to a basis, a type $(0, 2)$ tensor corresponds to a matrix $M$ with entries $M_{i j} = T_{i j}$ , and under a change of basis with matrix $A$ , this matrix transforms by congruence,

\tilde{M} = A^{T} M A

rather than by the similarity transformation used for type $(1, 1)$ tensors. This distinction in transformation behavior is the algebraic reason a bilinear form cannot, in general, be identified with a linear operator without an auxiliary choice of structure, such as a fixed reference metric, to convert between the two.

Distinction from Type (2, 0) and Type (1, 1)

Type $(0, 2)$ tensors, along with type $(2, 0)$ and type $(1, 1)$ tensors, are the three structurally distinct kinds of order-two tensor. Type $(0, 2)$ tensors act on pairs of vectors and transform by congruence; type $(2, 0)$ tensors act on pairs of covectors; and type $(1, 1)$ tensors act as linear operators, transforming by similarity. Distinguishing these three cases correctly is essential in any setting, such as differential geometry or continuum mechanics, where several order-two tensors of genuinely different types coexist and must not be conflated merely because each is represented, once a basis is fixed, by an ordinary two-index array.

Role Within Tensor Algebra

Type $(0, 2)$ tensors occupy the purely covariant, order-two slot of the type-graded tensor algebra, generalizing single covectors to pairs of dual-space factors and providing the natural home for metric tensors, symplectic forms, and every other bilinear form on a vector space. Together with type $(2, 0)$ and type $(1, 1)$ tensors, it completes the classification of order-two tensors by type, and among the three, it is the case most directly responsible for equipping an otherwise unstructured vector space with the geometric notions of length, angle, and orientation.