Difference between revisions of "Smooth manifold"
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| − | A [[Smooth function|smooth function]] on a smooth {{n|manifold}}, {{M|M}}, is a function {{M|f:M\rightarrow\mathbb{R}^k}} | + | 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)}} 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. | + | {{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. | 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. | ||
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
