You are thinking in terms of long geodesics, but your question is local. The metric is defined point by point. To know the metric tensor at a point $A,$ it is only necessary to know the geodesic distances from $A$ to all points in an arbitrarily small ball (germ) around $A.$ As to your second question, $S$ would need to be a fairly large set, as on $C^\infty$ manifolds one can alter the metric in a tiny ball (disk in dimension 2) without this being noticeable from far away. It may be that your set $S$ could be a finite set of triangulations, such that every point is in the interior of at least one simplex from at least one triangulation. So the idea, and I am not sure, is a Riemannian metric in the interior of a simplex determined by all pairwise distances between points on the boundary? If so, recovering the metric from such information is seriously nontrivial. But proving or disproving uniqueness is a smaller problem.
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You are thinking in terms of long geodesics, but your question is local. The metric is defined point by point. To know the metric tensor at a point $A,$ it is only necessary to know the geodesic distances from $A$ to all points in an arbitrarily small ball (germ) around $A.$ As to your second question, $S$ would need to be a fairly large set, as on $C^\infty$ manifolds one can alter the metric in a tiny ball (disk in dimension 2) without this being noticeable from far away. |
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