Let $M$ be a compact manifold of unknown dimension embedded in $\mathbb{R}^n$ for some natural $n$. Let $S=\{x_1, x_2,..., x_k\}$ be a uniform random sample obtained from $M$. Let $\Gamma(S)$ be the set of all geodesics between points in $S$ on $M$. Suppose we are given some subset of $\Gamma(S)$ (perhaps finite), what can be said of $M$ from this information?
I realize that this question is rather broad - are there any results regarding such questions (or at least similar). I have seen some work regarding persistent homology and "point clouds"; however, this seems to be a rather different question.
Additional comments: I have toyed with the idea of projecting the points of $S$ via the maps,
$\rho_i(x_j)=(x_j^{(1)}, x_j^{(2)},...,x_j^{(i-1)}, 0, x_j^{(i+1)},..., x_j^{(n)})$
It seems to me that if we were to define $N_\epsilon(x_j)=\{x_l\in S|d(x_j, x_l)\le\epsilon\}$ and $|N_\epsilon(x_j)|$ as the density of $x_j$, we would find that the density of $rho_i(x_j)$ increases if there is curvature in the $i-th$ direction.
This construction is rather simple and does not address the information we have about the geodesics but I figured that it might help.