Ideas on Representing an Affine Connection

I think the following suffices.

Let A and B be two neighbors across a facet. B has just one vertex b that does not belong to the facet. Given an affine connection, the location of b may be expressed in A’s zbcc coordinate system. This is a geometric assertion, not an suggestion of how to compute these coordinates. This can be seen if we imagine the two neighbors embedded in a flat n-space.

Of course the situation between the two neighbors is symmetric and perhaps it would be good for each to carry the zbcc coordinates of the far vertex of each of its neighbors. Neither is it clear whether when this calculation should be carried out. It might be performed in a phase following morphogenesis, or as a ray crosses the facet, and then cached.

The metric provided by ds2 data for each edge suffices for this calculation, even, I think, for non-positive definite metrics.

When the missing vertices are the origins for the respective zbcc’s, then each expression describes the same vector, that from one missing vertex to the other, with opposite signs. The pictures below show the respective coordinates:

We introduce new notation here and resume this calculation.