Manifold and Metric: Answers to Your Questions

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A manifold does not necessarily require a metric, as smooth manifolds can exist without being endowed with a Riemannian or Lorentzian metric. There are various structures that can exist between a basic smooth manifold and one that has a defined metric. A smooth manifold without a metric tensor is simply referred to as a smooth manifold. Thus, it is possible for manifolds to exist without metrics, highlighting the flexibility in their mathematical definitions. Understanding these distinctions is crucial in the study of manifold theory.
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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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