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Suppose $M$ is a complete Riemannian manifold with very large injectivity radius (say larger than $100$) and $\left\lbrace x_i: i \in I\right\rbrace$ is a maximal $1$-separated subset of $M$.

Is diffeomorphism class of $M$ determined by the (possibly infinite) distance matrix $(d(x_i,x_j))_{i,j \in I}$?

Suppose now that we are only given the information of which points $x_i,x_j$ are at distance less than $2$. Is this enough to determine $M$ up to diffeomorphism?

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My knowledge of this subject is obsolete, but anyway here are some partial answers.

If you have a Ricci curvature bound in addition to the injectivity radius, then you can recover the diffeomorphism type, e.g. with harmonic coordinates, see M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102 (1990), no. 2, 429-445. But 100 should be replaced by some radius depending on the dimension and the Ricci curvature bound.

Correction. It is the constant 1 that would depend on the Ricci curvature bound. (I did the rescaling argument wrong.)

Without curvature bounds things get more complicated. It is very easy to recover homotopy type, In fact, you don't need injectivity radius, only a contractibility function: every ball of radius $r\le r_0$ is contractible within a ball of radius $f(r)$. Given this, you can construct homotopy equivalence from an almost isometry defined on a net: extend it step by step to skeletons of a cell decomposition, just make sure that $n$ iterations of $f$ (or maybe $2f$) do not grow bigger than your constant 100.

If the dimension is not 3 (and maybe for 3 as well, since the Poincare conjecture is now solved), you can recover homeomorphism type under a pre-compactness assumption (in your set-up, this pre-compactness boils down to something like that every ball of radius 2 is covered by a bounded number of balls of radius 1, and then your radius 100 depends on this number). This follows from the arguments in Grove-Petersen-Wu, Geometric finiteness theorems via controlled topology. Invent. Math. 99 (1990), no. 1, 205-213.

Correction. Again, the pre-compactness would translate into something more complicated after rescaling the injectivity radius to 1.

They also use only local contractibility function, and in this more general context diffeomorphism stability fails in dimension 4. Of course this does not answer your question since the injectivity radius assumption is so much stronger. I don't know how essential the pre-compactness is. They certainly need it for their result (which is finiteness of topology types) but perhaps it may be relaxed if you only need stability.

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  • $\begingroup$ Since there is no upper diameter bound, the precompactness will be in pointed topology, so how would it capture the topological type of the manifold? $\endgroup$ Dec 11, 2012 at 23:49
  • $\begingroup$ @Igor: You are right, one has to assume that the diameter is bounded or check that the proof works for pointed topology. I have not checked this because they use some topology which I don't know well. $\endgroup$ Dec 12, 2012 at 9:11
  • $\begingroup$ Thanks for the references! I'm interested mostly in the non-compact case but this seems like a good starting point. $\endgroup$ Dec 12, 2012 at 16:33

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