One might be tempted, by analogy with the special-relativistic case, to require that the curves be autoparallels of the
inertia-gravitational connection (geodesics of the metric), but this is not necessary in theory, and in practice often not
helpful: autoparallels in a gravitational field tend to form caustics. Einstein’s term for a similar construction
was “mollusk of reference” (Einstein, 1952); but it was based on a coordinate system rather than a geometric
construction.