### Rigid Bodies, Flat Space, Relativity and Einstein’s Summation Convention

Einstein introduced his summation convention saying that when the same script appeared just twice in a term, once as a superscript and once as a subscript, then a summation sign for that script running over the four dimensions was to be imagined before the term.
In the context of a curved manifold with a metric tensor, adherence to this index pattern avoids many bugs.
In a flat space things are simpler.
For a standard Cartesian coördinate system the metric tensor is g_{ij} = δ_{ij} which is Kronecker’s delta.
An expression like x^{jk}g_{jk} in the context of a curved manifold becomes x^{jk}δ_{jk} in a flat space with conventional Cartesian coördinates.
In turn x^{jk}δ_{jk} = x^{kk} but at the cost of abandoning the rule of upper and lower scripts.
The upper-lower rule is not needed when g_{ij} = δ_{ij}.

The upper and lower rule ensured that only vectors from a space and its dual were “multiplied”.
The distinction is unnecessary in Cartesian coördinates.