Difference between revisions of "Smooth manifold"
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(Created page with " ==Definition== A ''smooth manifold'' is<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> a pair {{M|(M,\mathcal{A})}} where {{M|M}} is a Topological...") |
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We may now talk about "smooth manifolds" | We may now talk about "smooth manifolds" | ||
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| + | ==Notes== | ||
| + | * A [[Topological manifold|topological manifold]] may have many different potential [[Smooth structure|smooth structures]] it can be coupled with to create a smooth manifold. | ||
| + | * There do exist [[Topological manifold|topological manifolds]] that admit no smooth structures at all | ||
| + | ** First example was a compact 10 dimensional manifold found in 1960 by Michel Kervaire<ref>Ker60 in Introduction to smooth manifolds - John M Lee - Second Edition</ref> | ||
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| + | ==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: | ||
| + | * [[Smooth structure determined by an atlas]] | ||
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| + | ==Other names== | ||
| + | * Smooth manifold structure | ||
| + | * Differentiable manifold structure | ||
| + | * {{M|C^\infty}} manifold structure | ||
==See also== | ==See also== | ||
Revision as of 07:15, 7 April 2015
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
