4

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)?

Joseph Zambrano
  • 45
  • 1
    What sort of assumptions would you make on your subset of geodesics? Do you have any specific example (say, all shortest length geodesics)? – Marco Golla Jan 03 '15 at 22:57
  • 1
    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. – Ryan Budney Jan 03 '15 at 23:04
  • I will first apologize for stating this question very loosely. To be quite honest I do not know what assumptions would be particularly useful. My intuition tells me that it would be beneficial to have, for example, some information about geodesic loops - hopefully telling us something about the homology/homotopy. But I should have been clear in stating that I don't have anything particular in mind. – Joseph Zambrano Jan 03 '15 at 23:06
  • Ryan Budney - if possible, could you elaborate on your definition of "short geodesic" and what obstructions we might face in recovering a PL structure? – Joseph Zambrano Jan 04 '15 at 00:56
  • 1
    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. – Manfred Weis Jan 04 '15 at 06:53
  • 2
    I hope you are familiar with the work of niyogi, smale and weinberger (http://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. – Vidit Nanda Jan 04 '15 at 09:06
  • 2
    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. – Joseph O'Rourke Jan 04 '15 at 13:36
  • Thank you everyone, this has been incredible helpful. Vidit Nanda - yes, my question was actually inspired by a very brief look at their work, I really must take the time to thoroughly go through it. Joseph O'Rourke - wonderful links. – Joseph Zambrano Jan 04 '15 at 18:59

0 Answers0