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I am trying to understand the relationship between a simplicial complex and its corresponding Riemann manifold. Reading this post I understand that a simplicial complex is homeomorphic to a corresponding manifold, though I am not sure under what conditions this homeomorphism property exists or fails.

So the first question is:

  1. Does each simplicial complex correspond or map to some underlying manifold--in my case a Riemann manifold, or is the mapping only up to some homeomorphism?

But the real question is as follows.

  1. If a simplicial complex is homeomorphic to some corresponding manifold, then do geodesics between points on a simplicial complex correspond to geodesics on the underlying manifold? In other words, does the simplicial complex preserve distances vis-a-vis the corresponding manifold. Perhaps the distances are preserved up to some homeomorphism as well.

Any clarification would be appreciated. Let me know if I any additional details is needed for this question. Differential topology is not my forte, as I come from a stats background.

krishnab
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    I'm not really knowledgeable about differential geometry, but I suspect there is no relation. A triangulation of a manifold simply gives a homeomorphism between it and simplicial complex. Homeomorphism does not necessarily preserve metric properties like geodesics. Of course, we can use the chosen homeomorphism to define a new metric on simplicial complex. But I would "naturally" think of the individual faces of a simplicial complex as equipped with a flat Euclidean metric. And that metric is not the same as that of the manifold. – Jyrki Lahtonen Dec 03 '20 at 05:15
  • (cont'd) For example, is it not possible to define the Gaussian curvature of a sphere in terms of its geodesics? Making the metric locally incompatible with that of a flat face of a simplex. – Jyrki Lahtonen Dec 03 '20 at 05:17
  • @JyrkiLahtonen thanks for the valuable insight here. If the idea of the simplicial complex is to represent some notation of topology of a manifold--the triangulation, as it were, then perhaps using the natural metric on the complex is just as useful as the original metric on the manifold. I am looking at a question related to data analysis, and we can only recover/approximate the underlying manifold from a point cloud up to the tolerance of some numerical method. – krishnab Dec 03 '20 at 06:05
  • (cont'd) So perhaps I can just use the natural Euclidean metric on the complex to represent the distance between points, without needing to resort to the metric of the underlying manifold. – krishnab Dec 03 '20 at 06:06
  • Hard to tell without knowing the details. Using the metric on a simplicial complex may distort your metric. On the other hand, if the topology is correct, then the distortion should be bounded (compactness), and the Euclidean metric is still useful for your data analysis? A risk that occurs to me is that relatively small changes in the data set may lead to somewhat critical changes in the "estimated" topology, and that might have a bigger impact on the metric. Of course, I know nothing about your application, so cannot tell whether such a risk is relevant. – Jyrki Lahtonen Dec 03 '20 at 06:30

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