Difference between revisions of "Smooth manifold"

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'''Note: ''' It's worth looking at [[Motivation for smooth manifolds]]
  
 
==Definition==
 
==Definition==
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We may now talk about "smooth manifolds"
 
We may now talk about "smooth manifolds"
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==Quick guide==
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===Smoothly compatible charts===
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(See [[Smoothly compatible charts|smoothly compatible charts]]) - Two charts are smoothly compatible if the intersections of their domains is empty, or there is a [[Diffeomorphism|diffeomorphism]] between their domains. That is given two charts {{M|(A,\alpha)}} and {{M|(B,\beta)}} that:
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* {{M|1=A\cap B=\emptyset}} or
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* {{M|\beta\circ\alpha^{-1}:\alpha(A\cap B)\rightarrow\beta(A\cap B)}} is a [[Diffeomorphism|diffeomorphism]]
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===Smooth Atlas===
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A [[Smooth atlas|smooth atlas]] is an [[Atlas|atlas]] where every chart in the atlas, {{M|\mathcal{A} }}, is smoothly compatible with all the other charts in {{M|\mathcal{A} }}
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===Smooth function===
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A [[Smooth function|smooth function]] on a [[Smooth manifold|smooth {{n|manifold}}]], {{M|(M,\mathcal{A})}}, is a function<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> {{M|f:M\rightarrow\mathbb{R}^k}} that satisfies:
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{{M|\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} }} such that {{M|f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k }} is [[Smooth|smooth]] in the usual sense, of having continuous partial derivatives of all orders.
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Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so {{M|f}} is still smooth.
  
 
==Notes==
 
==Notes==
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* [[Smooth]]
 
* [[Smooth]]
 
* [[Smoothly compatible charts]]
 
* [[Smoothly compatible charts]]
 
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* [[Motivation for smooth manifolds]]
 
==References==
 
==References==
 
<references/>
 
<references/>
  
 
{{Definition|Manifolds}}
 
{{Definition|Manifolds}}

Latest revision as of 21:09, 12 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"

Quick guide

Smoothly compatible charts

(See smoothly compatible charts) - Two charts are smoothly compatible if the intersections of their domains is empty, or there is a diffeomorphism between their domains. That is given two charts [ilmath](A,\alpha)[/ilmath] and [ilmath](B,\beta)[/ilmath] that:

  • [ilmath]A\cap B=\emptyset[/ilmath] or
  • [ilmath]\beta\circ\alpha^{-1}:\alpha(A\cap B)\rightarrow\beta(A\cap B)[/ilmath] is a diffeomorphism

Smooth Atlas

A smooth atlas is an atlas where every chart in the atlas, [ilmath]\mathcal{A} [/ilmath], is smoothly compatible with all the other charts in [ilmath]\mathcal{A} [/ilmath]

Smooth function

A smooth function on a smooth [ilmath]n[/ilmath]-manifold, [ilmath](M,\mathcal{A})[/ilmath], is a function[2] [ilmath]f:M\rightarrow\mathbb{R}^k[/ilmath] that satisfies:

[ilmath]\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} [/ilmath] such that [ilmath]f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k [/ilmath] is smooth in the usual sense, of having continuous partial derivatives of all orders.

Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so [ilmath]f[/ilmath] is still smooth.

Notes

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

See also

References

  1. Introduction to smooth manifolds - John M Lee - Second Edition
  2. Introduction to smooth manifolds - John M Lee - Second Edition
  3. Ker60 in Introduction to smooth manifolds - John M Lee - Second Edition