Smooth structure

Definition

A smooth structure is a maximally smooth atlas, which recall is just a smooth atlas where every chart smoothly compatible with a chart in the atlas is already in the atlas, or a smooth atlas which is not properly contained in any larger smooth atlas

Motivation

We wish to define "smooth" functions on manifolds, eg $f:M\rightarrow\mathbb{R}$ is smooth if and only if $f\circ\varphi^{-1}$ is smooth in the usual sense (calculus, see Smooth) for each chart in the atlas. However as this example shows there are many smooth atlases giving the same "smooth structure"

Example: consider the two smooth atlases:

• $$\mathcal{A}_1=\{(\mathbb{R}^n,\text{Id}_{\mathcal{R}^n}\}$$
• $$\mathcal{A}_2=\{(B_1(x),\text{Id}_{B_1(x)})|x\in\mathbb{R}^n\}$$ (where $B_r(x)$ denotes an Open ball)

Clearly a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is Smooth with respect to either atlas if and only if it is Smooth in the usual sense.

Option 1

We could define a smooth structure as an Equivalence class of smooth atlases however it is far easier to use the second option

Option 2

Define the notion of a maximally smooth atlas

Other names

• Differentiable structure
• $C^\infty$ structure

See also

• Smooth manifold