Difference between revisions of "Smooth manifold"
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Revision as of 22:19, 9 April 2015
Note: It's worth looking at Motivation for smooth manifolds
Definition
A smooth manifold is[1] a pair [ilmath](M,\mathcal{A})[/ilmath] where [ilmath]M[/ilmath] is a topological [ilmath]n[/ilmath]-manifold and [ilmath]\mathcal{A} [/ilmath] is a smooth structure on [ilmath]M[/ilmath]
We may now talk about "smooth manifolds"
Notes
- A topological manifold may have many different potential smooth structures it can be coupled with to create a smooth manifold.
- There do exist topological manifolds that admit no smooth structures at all
- First example was a compact 10 dimensional manifold found in 1960 by Michel Kervaire[2]
Specifying smooth atlases
Because of the huge number of charts that'd be in a smooth structure there's little point in even trying to explicitly define one, see:
Other names
- Smooth manifold structure
- Differentiable manifold structure
- [ilmath]C^\infty[/ilmath] manifold structure
