# Sampling from a Manifold

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)?

• What sort of assumptions would you make on your subset of geodesics? Do you have any specific example (say, all shortest length geodesics)? Commented Jan 3, 2015 at 22:57
• If you chose "short" geodesics in an appropriate way you should be able to derive some kind of simplicial approximation to the manifold. You'd have to put strong restrictions to get the PL type, but you should be able to recover the homotopy type if you choose your criterion carefully. Commented Jan 3, 2015 at 23:04
• examples of canonical subsets of the geodesics could be those that constitute to: a minimum-weight matching, a minimum-weight apanning tree, a delaunay triangulation, or the shortest tour through all points of the sample. Commented Jan 4, 2015 at 6:53
• I hope you are familiar with the work of niyogi, smale and weinberger (people.cs.uchicago.edu/~niyogi/papersps/NiySmaWeiHom.pdf) which assumes that the manifold in question has been embedded in euclidean space and provides bounds on the sample size in terms of the injectivity radius to extract homotopy type with high confidence. Commented Jan 4, 2015 at 9:06
• Related questions: "Probing a manifold with geodesics," and "Can one recover a metric from geodesics?." The answer to the latter question is No. But see Robert Bryant's Yes answer to the former question, Yes under certain assumptions. Commented Jan 4, 2015 at 13:36