Smooth manifold

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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