Difference between revisions of "Smooth manifold"
m |
m (→Smooth function) |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | '''Note: ''' It's worth looking at [[Motivation for smooth manifolds]] | ||
==Definition== | ==Definition== | ||
| Line 4: | Line 5: | ||
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== | ==Notes== | ||
| Line 24: | 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
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
- 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
- [ilmath]C^\infty[/ilmath] manifold structure
