Smooth manifold

Note: It's worth looking at Motivation for smooth manifolds

Definition

A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological $n$-manifold and $\mathcal{A}$ 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 $(A,\alpha)$ and $(B,\beta)$ that:

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

Smooth Atlas

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

Smooth function

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

$\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A}$ such that $f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k$ 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

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
• $C^\infty$ manifold structure

See also

• Manifold
• Smooth structure
• Smooth
• Smoothly compatible charts
• Motivation for smooth manifolds