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

wonderingmd

Background:

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?

bcrowell

There are metric and non-metric theories of gravity. Metric theories may also have other fields besides the metric, as in Brans-Dicke gravity and other tensor-scalar theories. MTW discusses the reasons for preferring metric theories in section 38.4. In section 39.2, they state that "All the theories known to be viable in 1973 are metric, except Cartan's."

wonderingmd

There is a time metric, we experience it directly, and if there is time-passage variation, defacto there is a 'non-trivial' time-metric, but a more than one-dimensional context is necessary to give it meaning in terms of 'curvature', so could one indeed say it is a 'non-trivial time-metric' even if there is no full space-time metric?

And re the other issues: 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? (it is a way of relating SR time-dilation and GR time-passage-variation)