Abstract index notation vs Ricci Calculus I have come accross some comparison between the abstract index notation and Ricci calculus as it pertains to contraction and what I find is:
The former (abstract notation) indicates that a basis-independent trace operation being applied, which reduces to the aforementioned summation whenever a specific basis is fixed; the latter (Ricci calculus) construes contraction as true summation with numerical indexes and, correspondingly, with a given coordinate system.  In the Ricci calculus, a contraction indicates a literal summation. Since this requires numbers, it also requires a coordinate system to be chosen.  Really, the abstract index notation is nothing more than the observation that almost all of the Ricci calculus remains intact if one does not choose a basis. There's a great deal of meaning in the structure of the index expressions which is not basis dependent.
My question is very simple: What kind of meaning in the structure is not basis dependent? Can you elaborate on that?
Thanks in advance

A simple example to illustrate the point is an expression of the form $t^a_{\;a}$ where $t^a_{\;b}$ is a tensor (or tensor field) of type (1,1), vulgarly known as an endomorphism.  The expression $t^a_{\;a}$ is interpreted in the usual index notation as the sum of the "diagonal" elements and taken literally is dependent on the basis.  When interpreted as contraction in the abstract index notation, it results in a scalar (or scalar function) without ever having to choose a basis.
