manifold and metric

princeton118

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

Michael Mozina

Bob. :)

Chris Hillman

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.

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