At some point between high school and college we first make the transition between Euclidean (or synthetic) geometry and co-ordinate (or analytic) geometry. Later, during graduate studies we are introduced to differential geometry of many dimensions. The justification given in the first instance is that coordinates are a natural outcome of the axioms of Euclidean geometry; and in the second case because Riemannian geometry is much more general than axiomatic non-Euclidean geometry. In this expository account we examine these two justifications.
First of all due to the existence of non-Euclidean geometries one should take as the starting point not the usual axioms of Euclidean geometry but the ``local'' axioms proposed by Veblen [5], Hilbert [4] and others. These are satisfied by a small convex region within any of the axiomatic geometries--Euclidean or not. We will then follow (in Section 2) the exposition of Coxeter (see [2]) to show that any such geometry can be embedded within Euclidean space (of dimension 3). Thus there is a choice of coordinates for any such geometry--a justification for the first step above.
Now it is clear that there are many possible choices of coordinates. However, from the various physical and other applications it becomes clear that we must restrict our attention to those changes of coordinates which are differentiable with respect to one another. The search is then on for ``differential covariants'' or quantities that change in some systematic fashion with such a change of coordinates. An important such invariant is the Riemannian curvature and its other face, the sectional curvature (see Section 3).
We now examine the geometry constructed by axiomatic means to see what the sectional curvatures of this geometry can be. A combination of certain results of Schur, Cartan and Hadamard shows us that indeed we only obtain the ``classical'' geometries--flat Euclidean space, hyperbolic space (of Lobachevsky and Bolyai) and elliptic space (of Poncelet and Riemann)--the geometries of constant sectional curvature. Thus Riemannian geometry does lead to more general non-Euclidean geometries than those that can be constructed by axiomatic means. This gives a justification for the second step away from synthetic geometry.
In a final summarising section we suggest some additional reading and also point out some other interesting ways of constructing geometries that have not been covered here.