7.5 Tensor Component Index Structure

Tensor Component Index Structure is the complete specification of how many indices a tensor's components carry, which of those indices are upper and which are lower, and in what order they are arranged, forming the schema that governs how the tensor's array representation is built and interpreted.

Definition and Scope

The Three Defining Questions

An index structure answers three questions for a given tensor: how many indices are present in total, how many of those are upper, contravariant, indices versus lower, covariant, indices, and in what sequence the indices are listed. For a tensor of type ((p,q)), the index structure is summarized as

with (p) upper slots and (q) lower slots, each index ranging independently over the dimension of the underlying vector space.

From No Indices to Many

Index structure spans a full range of complexity, from the empty index structure of a scalar, through the single-index structure of a vector or covector, to the many-index structures of higher-rank tensors such as the curvature tensors of differential geometry, which commonly carry four or more indices in a fixed mixed arrangement of upper and lower slots.

Structural Properties

Variance Pattern

The variance pattern, the specific sequence of upper and lower designations, is part of the index structure and is not interchangeable without changing the tensor's type. A ((2,0)) tensor, a ((1,1)) tensor, and a ((0,2)) tensor can all be built over the same vector space with the same total index count, yet each has a distinct index structure with different transformation behavior under a change of basis.

Symmetry as a Constraint on Index Structure

Symmetry and antisymmetry properties are stated relative to a tensor's index structure, applying only to pairs of indices sharing the same variance. Describing a tensor's index structure completely therefore includes noting any symmetry constraints among its upper indices, among its lower indices, or, where a metric permits comparison, across the two.

Index Structure Under Combination

Combining tensors through operations such as the tensor product or contraction produces predictable changes to index structure: a tensor product concatenates the index structures of its two factors, while a contraction removes one upper and one lower index from the combined structure, always preserving the underlying rule that upper and lower index counts change only through such paired removals or additions.

Role Within Tensor Algebra

Determining Which Operations Are Valid

Index structure governs which operations between two tensors are algebraically valid before any numerical computation is performed: contraction requires an upper index in one tensor and a lower index in another with matching dimension, and addition requires two tensors to share an identical index structure entirely, both conditions checked directly against the stated index structure of the operands.

Foundation for Notation and Storage

Every notational and computational convention discussed for tensors, from the transformation law under change of basis to the layout of the component array in memory, is expressed in terms of the index structure, making it the single specification from which the tensor's algebraic behavior and its practical representation both follow.