Newtonian Gravity In Curved Space

There are several puzzling things about Newtonian gravity in a 
closed universe, not limited to the perennial question about
whether Gauss's law applies in such a context.  On a very basic 
level, any force-at-a-distance theory gets into trouble if 
"distance" is not unique.  Any spatial geometry that allows local 
curvature will lead to local ambiguities.  Even global curvature 
seems to imply that the laws would not really be inverse-square.

For example, consider two objects in a spherical universe of 
circumference C.  If the onjects are separated by a distance r
then they are also separated by a distance C-r in the opposite
direction, so naively applying the inverse square formulation
would give
                        1          1
                F  ~  -----  -  -------
                       r^2      (C-r)^2

I remember once computing the value of C that would give the correct 
precession of Mercury's orbit.  As I recall, it worked out to only 
about 7 times the radius of Pluto's orbit, which would make for a 
pretty cramped universe.  Of course, the above formula isn't really 
complete, because the objects are also separated by the distances 
r+C, r+2C, r+3C,... and also r-C, r-2C, r-3C,...so the actual inverse 
square force law would be in infinite series.

Even more puzzling is an inverse-square force law in a space with
a cylindrical dimension.  Consider two points on a cylinder of 
circumference C.  If the two points are at the same angular position 
but separated by an axial distance of R, then the points are obviously 
separated by a geodesic of length R.  However, they are also separated 
by a geodesic of length sqrt(R^2 + C^2) that loops once around the 
cylinder.  In fact, they are separated by infinitely many geodesics
of length sqrt(R^2 + nC^2), n=0,1,2,..., where n denotes the number
of loops around the cylindrical dimension.  This again gives an infinite 
series for the complete force law, but now the circumference C need 
not be large.  You can imagine Kaluza-Klein type universes with curled 
up dimensions that would lead to very strange force laws.  I suppose 
you could try to find a "first-order" law that would yield an infinite 
series whose sum approaches an inverse-sqaure law.

Anyway, I think this all goes to show that once you depart from perfect 
Euclidean space (and absolute time), it's hard to make much sense out 
of action-at-a-distance force laws.  This is especially true when the 
plane of simultaniety is not unique, since that requires you to look 
for action-at-a-distance laws that remain satisfied for any possible 
choice of reference frame.  I imagine this severely constrains the 
possibilities, although I recall reading once about an action-at-a-
distance theory that was consistent with Minkowski space-time (locally).