Manifold and Metric: Answers to Your Questions

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SUMMARY

A manifold does not necessarily require a metric; smooth manifolds can exist without a Riemannian or Lorentzian metric. There are various intermediate structures between a smooth manifold and a Riemannian or Lorentzian manifold. Specifically, any smooth manifold lacking a metric tensor falls into this category, demonstrating that non-metric manifolds are indeed valid in mathematical theory.

PREREQUISITES
  • Understanding of smooth manifolds
  • Familiarity with Riemannian geometry
  • Knowledge of Lorentzian geometry
  • Basic concepts of manifold theory
NEXT STEPS
  • Research the differences between smooth manifolds and Riemannian manifolds
  • Explore the concept of metric tensors in Riemannian and Lorentzian geometry
  • Study intermediate structures in manifold theory
  • Investigate examples of non-metric manifolds in mathematical literature
USEFUL FOR

Mathematicians, students of differential geometry, and anyone interested in the theoretical foundations of manifolds and their properties.

princeton118
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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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princeton118 said:
Does a manifold necessarily have a metric?
Does a manifold without metric exist? If it exists, what is its name?

Bob. :)
 
princeton118 said:
Does a manifold necessarily have a metric?

Well, you posted this in the General Astronomy forum but I will interpret your question to concern the theory of manifolds in mathematics. With that assumption, no, in general, smooth manifolds need not be provided with any Riemannian (or Lorentzian) metric. There are in fact several intermediate levels of structure between the basic notion of a smooth manifold and the notion of a Riemannian (or Lorentzian) manifold.

princeton118 said:
Does a manifold without metric exist?

Any smooth manifold which has not been provided with a metric tensor in the sense of Riemannian (or Lorentzian) geometry.
 

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