Given a Lorentzian manifold and metric tensor, "$( M, g )$", the corresponding causal relations between its elements (events) may be derived; i.e. for every pair (in general) of distinct events in set $M$ an assignment is obtained whether it is timelike separated, or lightlike separated, or neither (spacelike separated).
In turn, I'd like to better understand whether causal separation relations, given abstractly as "$( M, s )$", allow to characterize the corresponding Lorentzian manifold/metric. As an exemplary and surely relevant characteristic (cmp. answer here) let's consider whether the Riemann curvature tensor vanishes, or not, at each event of the whole set $M$ (or perhaps suitable subsets of $M$).
Are there particular causal separation relations which would be indicative, or counter-indicative, of the Riemann curvature tensor vanishing at all events of set $M$ (or if this may simplify considerations: at all events of a chart of the manifold); or on some subset of $M$?
To put my question still more concretely, consider as possible illustration of "counter-indication":
(a)
Can any chart of a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor (or, at least, a whole such manifold) contain [Edit in consideration of 1st comment (by twistor59): -- the Riemann curvature tensor vanish at least in one event of a 3+1 dimensional Lorentzian manifold if each of its charts contains -- ]
fifteen events (conveniently organized as five triples):
$A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q\,\,\,\,$, and $\,\,\,\, U, V, W$,
where (to specify the causal separation relations among all corresponding one-hundred-and-five event pairs):
$s[ A, B ]$ and $s[ A, C ]$ and $s[ B, C ]$ are timelike,
$s[ F, G ]$ and $s[ F, H ]$ and $s[ G, H ]$ are timelike,
$s[ J, K ]$ and $s[ J, L ]$ and $s[ K, L ]$ are timelike,
$s[ N, P ]$ and $s[ N, Q ]$ and $s[ P, Q ]$ are timelike,
$s[ U, V ]$ and $s[ U, W ]$ and $s[ V, W ]$ are timelike,$s[ A, G ]$ and $s[ G, C ]$ and $s[ A, K ]$ and $s[ K, C ]$ and
$s[ A, P ]$ and $s[ P, C ]$ and $s[ A, V ]$ and $s[ V, C ]$ are lightlike,$s[ F, B ]$ and $s[ B, H ]$ and $s[ F, K ]$ and $s[ K, H ]$ and
$s[ F, P ]$ and $s[ P, H ]$ and $s[ F, V ]$ and $s[ V, H ]$ are lightlike,$s[ J, B ]$ and $s[ B, L ]$ and $s[ J, G ]$ and $s[ G, L ]$ and
$s[ J, P ]$ and $s[ P, L ]$ and $s[ J, V ]$ and $s[ V, L ]$ are lightlike,$s[ N, B ]$ and $s[ B, Q ]$ and $s[ N, G ]$ and $s[ G, Q ]$ and
$s[ N, K ]$ and $s[ K, Q ]$ and $s[ N, V ]$ and $s[ V, Q ]$ are lightlike,$s[ U, B ]$ and $s[ B, W ]$ and $s[ U, G ]$ and $s[ G, W ]$ and
$s[ U, K ]$ and $s[ K, W ]$ and $s[ U, P ]$ and $s[ P, W ]$ are lightlike,the separations of all ten pairs among the events $A, F, J, N, U$ are spacelike,
the separations of all ten pairs among the events $B, G, K, P, V$ are spacelike,
the separations of all ten pairs among the events $C, H, L, Q, W$ are spacelike, and finallythe separations of all twenty remaining event pairs are timelike
?
Conversely, consider as possible illustration of "indication":
(b)
Is there a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor (or, at least, one of its charts) which doesn't [Edit in consideration of 1st comment (by twistor59): -- nowhere vanishing Riemann curvature tensor such that all of its charts -- ] contain
twenty-four events, conveniently organized as
four triples ($A, B, C$; $\,\,\,\, F, G, H$; $\,\,\,\, J, K, L$; $\,\,\,\, N, P, Q$) and
six pairs ($D, E$; $\,\,\,\, S, T$; $\,\,\,\, U, V$; $\,\,\,\, W, X$; $\,\,\,\, Y, Z$; $\,\,\,\, {\it\unicode{xA3}}, {\it\unicode{x20AC}\,}$),
where (again explicitly, please bear with me$\, \!^*$):
the sixty-six separations among the twelve events belonging to the four triples are exactly as in question part (a),
each of the six pairs is timelike separated,
the separations of all fifteen pairs among the events $D, S, U, W, Y, {\it\unicode{xA3}}$ are spacelike,
the separations of all fifteen pairs among the events $E, T, V, X, Z, {\it\unicode{x20AC}\,}$ are spacelike,$s[ D, {\it\unicode{x20AC}\,} ]$ and $s[ S, Z ]$ and $s[ U, X ]$ are spacelike,
$s[ E, {\it\unicode{xA3}} ]$ and $s[ T, Y ]$ and $s[ V, W ]$ are spacelike,$s[ A, {\it\unicode{xA3}} ]$ and $s[ A, Y ]$ and $s[ A, W ]$ are spacelike,
$s[ A, {\it\unicode{x20AC}\,} ]$ and $s[ A, Z ]$ and $s[ A, X ]$ are timelike,
$s[ A, E ]$ and $s[ A, T ]$ and $s[ A, V ]$ are timelike,$s[ C, {\it\unicode{x20AC}\,} ]$ and $s[ C, Z ]$ and $s[ C, X ]$ are spacelike,
$s[ C, {\it\unicode{xA3}} ]$ and $s[ C, Y ]$ and $s[ C, W ]$ are timelike,
$s[ C, D ]$ and $s[ C, S ]$ and $s[ C, U ]$ are timelike,$s[ F, {\it\unicode{xA3}} ]$ and $s[ F, D ]$ and $s[ F, S ]$ are spacelike,
$s[ F, {\it\unicode{x20AC}\,} ]$ and $s[ F, E ]$ and $s[ F, T ]$ are timelike,
$s[ F, V ]$ and $s[ F, X ]$ and $s[ F, Z ]$ are timelike,$s[ H, {\it\unicode{x20AC}\,} ]$ and $s[ H, E ]$ and $s[ H, T ]$ are spacelike,
$s[ H, {\it\unicode{xA3}} ]$ and $s[ H, D ]$ and $s[ H, S ]$ are timelike,
$s[ H, U ]$ and $s[ H, W ]$ and $s[ H, Y ]$ are timelike,$s[ J, D ]$ and $s[ J, U ]$ and $s[ J, Y ]$ are spacelike,
$s[ J, E ]$ and $s[ J, V ]$ and $s[ J, Z ]$ are timelike,
$s[ J, T ]$ and $s[ J, X ]$ and $s[ J, {\it\unicode{x20AC}\,} ]$ are timelike,$s[ L, E ]$ and $s[ L, V ]$ and $s[ L, Z ]$ are spacelike,
$s[ L, D ]$ and $s[ L, U ]$ and $s[ L, Y ]$ are timelike,
$s[ L, S ]$ and $s[ L, W ]$ and $s[ L, {\it\unicode{xA3}} ]$ are timelike,$s[ N, D ]$ and $s[ N, S ]$ and $s[ N, W ]$ are spacelike,
$s[ N, E ]$ and $s[ N, T ]$ and $s[ N, X ]$ are timelike,
$s[ N, V ]$ and $s[ N, Z ]$ and $s[ N, {\it\unicode{x20AC}\,} ]$ are timelike,$s[ Q, E ]$ and $s[ Q, T ]$ and $s[ Q, X ]$ are spacelike,
$s[ Q, D ]$ and $s[ Q, S ]$ and $s[ Q, W ]$ are timelike,
$s[ Q, U ]$ and $s[ Q, Y ]$ and $s[ Q, {\it\unicode{xA3}} ]$ are timelike, and finallythe separations of all ninety-six remaining event pairs are lightlike
?
(*: The two sets of causal separation relations stated explicitly in question part (a) and part (b) are of course not arbitrary, but have motivations that are somewhat outside the immediate scope of my question -- considering Lorentzian manifolds -- itself. It may nevertheless be helpful, if not overly suggestive, to attribute the relations of part (a) to "five participants, each finding coincident pings from the four others", and the relations of part (b) to "ten participants -- four as vertices of a regular tetrahedron and six as middles between these vertices -- pinging among each other".)
