Manifold & Metric: Does it Need a Metric?

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SUMMARY

A manifold does not necessarily require a global metric, as evidenced by the existence of non-metrizable manifolds like the Prüfer manifold. While every manifold is locally Euclidean and possesses a local metric, this does not guarantee a global distance metric between all points. The hierarchy of structures includes topological spaces, topological manifolds, differential manifolds, and Riemannian or Pseudo-Riemannian manifolds, each building upon the previous. Urysohn's metrization theorem is essential for determining whether a manifold is metrizable.

PREREQUISITES
  • Understanding of topological spaces and their properties
  • Familiarity with differential manifolds and their structures
  • Knowledge of Riemannian and Pseudo-Riemannian metrics
  • Awareness of Urysohn's metrization theorem
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  • Study Urysohn's metrization theorem in detail
  • Explore the properties of the Prüfer manifold
  • Learn about the distinctions between metric spaces and metric tensors
  • Investigate the implications of Lorentzian metrics in manifold theory
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Mathematicians, theoretical physicists, and students of geometry who are exploring the concepts of manifolds, metrics, and their applications in various fields of study.

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Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?
 
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Since a manifold is locally Euclidean, it must always have a local metric. However, it does not follow that there will be a "distance" between ANY two points and so there may not be a "global" metric.
 
the hierarchy is something like this. You take a set. Add a topology. It becomes a tpological space. Add an atlas, it becomes a topological manifold. Change for a differentiable atlas. It becomes a differential manifold. Add a Riemannian or Pseudo-Riemannian of Lorentzian or whatever metric and it becomes a Riemannian (resp. Pseudo-Riemannian, Lorentzian, whatever) manifold.
 
Topological manifolds (sets with a topology locally homeomorphic to Rn) do not necessarily admit a metric. There are then many non-metrizable manifolds, such as the Prüfer manifold. Urysohn's metrization theorem will let you know if your manifold is metrizable.
 
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Ditto slider142 about topological manifolds. I have a hunch the OP will need to know that "metric space" is not the same thing as the notion of "metric tensor" from Riemannian geometry, although they are certainly related. But "metric tensor" from Lorentzian geometry is not much like "metric" from "metric space"!

About smooth manifolds: you can give a smooth manifold additional structure, perhaps by defining a Riemannian or Lorentzian metric tensor. As Halls hinted, as per the fundamental local versus global distinction in manifold theory, even after defining a Riemannian or Lorentzian metric tensor, there will be multiple distinct notions of "distance in the large" which may or not correspond roughly to the notion of "metric" fro m "metric space". In particular, Lorentzian metrics get their topology from the (locally euclidean) topological manifold structure, not from the bundled indefinite bilinear form.

(I'm being a bit more sloppy than usual due to PF sluggishness.)
 

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