Does Euclidean space need a separate geometric substrate from its vector space structure?

[lavinia]

What if it s given a connection, wouldn't that count as geometry?

What if it s given a connection, wouldn't that count as geometry?

Yes - in a more general sense.

- One could have a connection that is compatible with a metric but is not a Levi_Civita connection

- One could have a connection on the tangent bundle that is not compatible with any metric. In this case I do not see how distance relations can be derived - but not sure. Still one has curvature and parallel translation,

This thread though seems to assume metric relations of some kind.

 

[lavinia]

@WWGD Here is Kervaire's original paper where he describes a manifold that does not admit any differentiable structure.

http://web.math.rochester.edu/people/faculty/doug/otherpapers/kervaire.pdf

 

[WWGD]

@WWGD Here is Kervaire's original paper where he describes a manifold that does not admit any differentiable structure.

http://web.math.rochester.edu/people/faculty/doug/otherpapers/kervaire.pdf

Excellent, Lavinia, thanks, will check it out asap.