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Smoothness is pure convenience. You can define manifolds on any ##C^n## level. It is only a requirement for the charts. Take a piece of paper, fold it and you have a continuous manifold which is not differentiable at all.

Topological spaces are rarely a group. Either choose a classical example like a Sierpi´nsky space, Cantor sets, or any other of the usual suspects.

Groups which are manifolds (not the other way around!) have originally been called continuous groups, in contrast to discrete or finite groups. They were studied in calculus of variations. Hence they needed a structure on which calculus can be performed. The least requirement is thus continuity, smoothness the requirement for lazybones who do not want to manage the degree of differentiability at each single step of the considerations. Luckily the major matrix groups are all smooth, as they are defined via polynomials.

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Science Advisor

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All of these questions are strange. If you know what the terms you reffer to mean, then the answers should be obvious. If you don't, the the answers are not what you need.

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A metric.What makes a space a geometric space?

And continuity makes a space a topological space.

And a multiplication makes a space a group.

And differentiability makes a space a manifold.

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Is a vector space a geometric space?What makes it or not a geometric space?But can a manifold not be a geometric space? I think that manifolds may not be differentiable.A metric.

And continuity makes a space a topological space.

And a multiplication makes a space a group.

And differentiability makes a space a manifold.

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No, but often it is. You do not need a metric in a vector space, but many have.Is a vector space a geometric space?

Still a metric: angles and distances.What makes it or not a geometric space?

Yes, i.e. locally no, since it has homeomorphic charts, globally yes, because you normally don't have only one chart.But can a manifold not be a geometric space?

Yes. As I already told you. Continuity is sufficient.I think that manifolds may not be differentiable.

As the questions are answered and we are circling around definitions, this thread is now closed. Please read about those definitions on Wikipedia, nLab, or any other dictionary site.

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