Combining all the results we see that Riemannian geometry is indeed of a
more general nature than axiomatic geometry--at least over real numbers
and in dimension strictly greater than 2. One can generalise further by
dropping the requirement of a Riemannian metric and retaining the linear
or affine connection *D*. Yet another generalisation is to consider fields
other than tensor fields (for example spinor fields) and consider derivations
on such fields.

At the same time a number of interesting questions remain if the field is not that of real numbers (in dimension at least 3) or if we have a planar geometry which does not satisfy Desargues axiom (which is almost obvious in dimension 3)--in the latter case we do not even have a skew-field associated with the geometry. These are the subject of study in Combinatorial Geometry and Group theory. The latter (group theory) points to another approach to geometry--that taken by Pieri, Klein and Lie--where the fundamental notion is that of a group of motions. A similar study to the one undertaken above ought to show that this leads to Lie groups and Symmetric spaces.