One of the several definitions of an affine space goes like this. Let $M$ be an arbitrary set whose elements are called points, let $\mathcal{V}$ be a vector space of dimension $n$, and let $\lambda:\mathcal{M}\times\mathcal{M}\to\mathcal{V}$ have the following properties:
For each $p$ in $M$ and each $\vec{v}$ in $\mathcal{V}$ there is a unique $q$ in $\mathcal{V}$ such that $\lambda(p,q)=\vec{v}$
$\lambda(p,0)=p$ for each $p$ in $M$
$\lambda(p,r)+\lambda(r,q)=\lambda(p,q)$
For classical and special relativitistic physics, an affine space seems to model the physical facts nicely, but not for general relativity. For the latter, we jump to manifolds with an enormous jump in complexity and variability from one author to another.
My question is this: Where does the definition of affine space fall short? I strongly suspect it is in Axiom 3 above, which is a kind of linearity assumption. Unfortunately, no author seems to tackle the transition; all launch full bore into manifold theory. Could someone provide a reference that does-one which discusses the manifold axioms from the point of view of physical phenomena?
Furthermore, can we truly separate the issue of affine versus nonaffine from that of the metric?
Added Later: Please try to understand what I am asking here. There are many treatments available which cover the formalism, but they merely postulate local coordinates. What I am looking for is a discussion of the physical reason for limiting them to local neighborhoods. If our basic set is affine we can establish global coordinates relative to an arbitrarily chosen origin. Thus, given a vector $\vec{v}$ for instance, we can describe a straight line by $\vec{x}=\vec{x}_0+\vec{v}t$ extending to infinity in both directions. Can we thus describe a straight line in a general manifold? If not, then why not?
This post imported from StackExchange Physics at 2014-12-12 22:30 (UTC), posted by SE-user Heaviside