In order to tie up the first two sections we need to have a notion
analogous to lines in a general Riemannian manifold--this is provided
by *geodesics* or energy minimising paths.

To a stationary observer placed on the manifold it would appear that a
body travelling along an energy minimising path is subject to no
acceleration. The translation of this into differential geometric terms
is
*D*_{X(t)}(*X*(*t*)) = 0 where *X*(*t*) is the tangent vector at time *t*.
By the theory of second order ordinary differential equations there is
a unique geodesic starting at a point *p* with initial velocity
for any choice of *p* and .

A result of Whitehead shows that for any point *p* there is a small
region *M* surrounding it so that there is a unique geodesic in *M*
joining any pair of points in *M*. The notion of between-ness is
defined by saying that *B* lies between *A* and *C* in *M* if *B* lies
on the (unique) geodesic joining *A* and *C*. Extending the geodesic
within *M* beyond *B* and before *A* gives us the ``line'' joining *A*
and *B*. We then easily check that Veblen's axioms of local geometry other
than those involving planes are satisfied. In particular we have
trouble verifying the Pasch axiom--numbered 7 in the
list of Veblen's axioms given in section 1.

Let us therefore make the *additional* assumption that these
axioms dealing with planes are satisfied; we will show that this
imposes a restriction on the curvature of the Riemannian manifold *M*,
which is satisfied if and only if *M* is a convex region in one of the
``classical'' geometries--Euclidean, Hyperbolic or Projective.

According to the results of section 1 we can choose coordinates on *M*
in such a way that the geodesics are mapped into lines. In terms of
these coordinates we see that a geodesic must be an accelerated path:

(*X*(*t*)) = *D*_{X(t)}(*X*(*t*)) - (*X*(*t*))(*X*(*t*)) = - (*X*(*t*))(*X*(*t*))

being the acceleration at time
()() = ^{ . }

for some scalar
depending on .
In addition we have the torsion-free condition and the linearity of
in each variable. It follows that
= < ,
()() = ( < (*p*), > ^{ . } + < (*p*), > ^{ . })

Thus we have
Let

(*X*, *Y*) =

where the right hand side does not depend on
Now we have
(*X*, *Y*)(*p*) = (*p*) depends only on the point *p*.
Thus we have

So that if

0 = *D*_{W}^{ . } < *U*, *U* > *V* - *D*_{V}^{ . } < *U*, *U* > *W*

Finally, since have assumed that we are in (at least) three dimensions
we can assume that