2.1. Properties of indices: type, character, state

1) There are five known types of index in xTensor` (other types could be added, but there is no user-defined way to do this):
    - Abstract index (AIndex). See subsection 2.2.
    - Basis index (BIndex), associated to some basis (coordinated or not) of vector fields. Described in xCoba`.
    - Component index (CIndex), also associated to some basis. Described in xCoba`.
    - Directional index (DIndex). See subsection 3.7.
    - Label index (LIndex). See subsection 3.8.
Collectively we refer to all five classes as "generalized indices" (GIndex). Additionally, we can have patterns (PIndex) for g-indices at those positions where g-indices are expected, but patterns are not considered as g-indices. In the following "indices" will mean "g-indices" unless we specify the type.

2) All indices have a "character", which can be either covariant (Down-index) or contravariant (Up-index).

3) Some indices can be used to represent contractions, following the Einstein convention: two repeated indices in the same tensor or tensor product, but each having a different character.  We call them "contractible" or "Einstein" indices (EIndex) and currently only abstract and basis indices are allowed to be contractible. (Note that having contractible indices with the same character is a syntactic error in xTensor`.) E-indices have a "state": they can be Free or contracted (aka Dummy). For completeness, the state of components, directions and labels is said to be always "Blocked", what in particular means that they can be repeated (i.e. truly repeated, not simply staggered).

4) Basis and component indices are known by part of the routines of xTensor` (for example the formatting routines, the index selectors, the canonicalizer or the constructors of rules), but used in the twin package xCoba`. From now on we shall not consider those two types of indices. See xCobaDoc.nb for further information.
5) There is a further property of an index, only used internally: its metric-state, saying whether an index can be raised or lowered using a given metric (not whether it has been or not actually raised already). This will be important in section 7.3.

Created by Mathematica  (May 16, 2008)