6.24.4 Tensor Mixed Index Notation

Tensor Mixed Index Notation is the notation used for tensors that carry both upper and lower indices simultaneously, such as T^{i}_{j} or T^{ik}_{jl}, in which the superscripts and subscripts together record a type (p, q) with both p > 0 and q > 0, and in which the relative ordering and grouping of the raised and lowered positions must be tracked carefully because it affects how contraction and other operations are correctly performed. This notation sits between the purely contravariant notation (all upper) and the purely covariant notation (all lower), and it is the notation required for the great majority of tensors that arise in practice, including linear maps and the Riemann curvature tensor.

Reading a Mixed Index Expression

Counting Each Kind Separately

In an expression such as T^{ik}_{jl}, the type is read by counting superscripts and subscripts independently: here there are two upper indices (i, k) and two lower indices (j, l), giving type (2, 2) and order 4. The mixed notation requires no new counting rule beyond what applies to purely upper or purely lower notation; it simply applies both counts to the same symbol.

Slot Position Versus Variance Category

A subtlety specific to mixed notation is that the physical left-to-right order of indices within their own row (all-upper row, all-lower row) may or may not correspond to a specific "slot" of the tensor, depending on convention. Some sources use a placeholder dot or space to indicate exact slot alignment, writing T^{i}_{~j}^{~k} to show that the j-slot sits between the two upper slots in the underlying multilinear map, rather than after both of them; this finer bookkeeping matters when a tensor is not symmetric and the specific slot order affects the value of a contraction.

Why Mixed Notation Requires Extra Care in Contraction

Only Opposite-Variance Pairs May Be Contracted

is a legitimate contraction — summing an upper index against a lower index of the same letter — producing a basis-independent scalar (or lower-order tensor) precisely because one factor of the transition matrix A cancels against one factor of its inverse. In mixed notation, this pairing must be read off correctly: contracting two indices that are both upper, or both lower, is not a valid tensor operation, even if the letters match, because no such cancellation occurs.

Tracking Free Versus Summed Indices

In a mixed expression involving several terms, some indices are repeated and summed (dummy indices) while others appear once per term and remain free; the mixed notation convention requires that the set of free upper indices and the set of free lower indices match exactly across every term of a valid tensor equation, which is the index-balance check made possible by consistently applying mixed notation throughout.

Diagram of a Mixed-Index Tensor

Common Tensors Written in Mixed Notation

The Identity Map and the Kronecker Delta

The identity linear map on V is a type (1,1) tensor, written δⁱⱼ, and is the simplest nontrivial mixed-index tensor: its components equal 1 when i = j and 0 otherwise, in every basis, making it the unique (1,1) tensor whose component array is unchanged by any change of basis.

The Riemann Curvature Tensor

In differential geometry, the Riemann curvature tensor is most commonly written in mixed form as R^{i}_{jkl}, type (1, 3), with the single upper index and three lower indices reflecting its origin as a map that takes three vectors (the lower slots) and returns a fourth vector (represented by the upper slot); correctly maintaining the mixed index notation here is essential, since lowering or raising any one of its indices with the metric produces a related but numerically and notationally distinct tensor, such as the fully covariant R_{ijkl}.