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" | ||
| + | |||
| + | ==Quick guide== | ||
| + | ===Smoothly compatible charts=== | ||
| + | (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: | ||
| + | * {{M|1=A\cap B=\emptyset}} or | ||
| + | * {{M|\beta\circ\alpha^{-1}:\alpha(A\cap B)\rightarrow\beta(A\cap B)}} is a [[Diffeomorphism|diffeomorphism]] | ||
| + | |||
| + | ===Smooth Atlas=== | ||
| + | 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} }} | ||
| + | |||
| + | ===Smooth function=== | ||
| + | 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: | ||
| + | |||
| + | {{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. | ||
| + | |||
| + | 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== | ||
| + | * 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> | ||
| + | |||
| + | ==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]] | ||
| + | |||
| + | ==Other names== | ||
| + | * Smooth manifold structure | ||
| + | * Differentiable manifold structure | ||
| + | * {{M|C^\infty}} manifold structure | ||
==See also== | ==See also== | ||
| Line 10: | Line 41: | ||
* [[Smooth]] | * [[Smooth]] | ||
* [[Smoothly compatible charts]] | * [[Smoothly compatible charts]] | ||
| − | + | * [[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
Contents
[hide]Definition
A smooth manifold is[1] a pair (M,A) where M is a topological n-manifold and A is a smooth structure on M
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 (A,α) and (B,β) that:
- A∩B=∅ or
- β∘α−1:α(A∩B)→β(A∩B) is a diffeomorphism
Smooth Atlas
A smooth atlas is an atlas where every chart in the atlas, A, is smoothly compatible with all the other charts in A
Smooth function
A smooth function on a smooth n-manifold, (M,A), is a function[2] f:M→Rk that satisfies:
∀p∈M ∃ (U,φ)∈A such that f∘φ−1⊆Rn→Rk 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 f is still smooth.
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[3]
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
- C∞ manifold structure
