Manifold and Metric: Answers to Your Questions

[princeton118]

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

 

[Michael Mozina]

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

Bob. :)

 

[Chris Hillman]

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.

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.