Suppose we were to obtain a uniform sample, $S=\{x_1,...,x_m\}$, of points on a closed Riemannian $n$-manifold $M$. Let $\Gamma(S)$ be the set of all geodesics between the points in $S$ and we are given some subset of $\Gamma(S)$. What sort of information can we obtain about $M$ (provided we make some assumptions about the subset of geodesics we are given)?

No. But see Robert Bryant'sYesanswer to the former question,Yesunder certain assumptions. $\endgroup$3more comments