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...
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...