Originally Posted by

**lesaid**
To keep the car aligned, I think you could simply ensure the front and back wheels remain parallel to and rotating at the same speed as each other at all times. Then as the car moves, the displacement of front and back of the car to the front and back of where it was a moment earlier, would stay the same, and from the (local) perspective of the driver, the car would always be parallel to where it was a moment earlier. On a perfectly spherical earth, no need for a plumb line - the car would inevitably remain pointing along a tangent to the surface.

Parallel transport is a way of finding the curvature intrinsic to a surface without having to view it from 'outside' the system. It can find the curvature at a particular point by considering parallel transport around a small closed path on that surface. If that path passes over a variety of arbitrary hills and valleys, we no longer have a consistent curvature, and perhaps need to consider a smaller path to enable the local curvature to be found on a scale where the curvature is consistent. That analysis would then reflect the curvature in the vicinity of the path, incorporating whatever local variation might be present.

If, by raising and lowering wheels, you managed to keep the car tangential to the earth's surface with constant orientation, ignoring local variations, and navigated your path in terms of spherical coordinates with a constant radius (i.e. ignoring the height variations and perceived distances along the undulating 'two dimensional' surface), I think (intuitively) that you would still end up with a valid result for the 'spherical' earth.