A trichotomy theorem for o-minimal structures

Let script M sign = 〈M, <, . . .〈 be a linearly ordered structure. We define script M sign to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitrary o-minimal script M sign can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which script M sign induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which script M sign induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods.