To state the context, I am familiar with the Hopf-Rinow theorem.
My request is three fold,
I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots of examples for this.
I want to know of techniques of proving and testing for geodesic incompleteness or completeness.
I want to know of methods of proving and testing for existence of maximal extension of curves.
{Confusingly in quite a bit of literature I have seen the adjective "inextensible" being used when actually they seem to want to mean "maximally extended". Is there some subtle point here that I am missing?}