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Math:

An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics.

A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature.

Given the existence of a connection with Riemann curvature there does not have to be a metric.

Physics: there is time-duration variation on stationary vertically-separated clocks on the surface a mass source (they are not in free fall, so the time is not an affine parameter).

Question: Is this time-effect necessarily part of the geometric structure of space-time, or could it be that it is separate, and there is no full space-time metric- does the existence of this type of effect necessarily imply/require the existence of a space-time metric? So if we measured such a time variation affect alone without even seeing gravitational acceleration or feeling its 'pull' we could know - if we knew enough math - that spacetime is a Riemanian manifold with a metric?

What type of non-purely-gravitational phenomena that we are familiar with in our universe could not exist in a universe with Riemann curvature outside a mass but without a space-time metric?

A string of vertically-separated clocks (separation d) fall freely past a stationary string of vertically-separated clocks (separation d), the times on each are compared, as are the time-lapses between meetings. What is observed?

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# Given Riemann & connection, does time variation nec'ly imply full spacetime metric?

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