Smooth structure

Builds on Topological manifold Atlas Homeomorphism Smooth function

Definition

Let $M$ be a topological manifold of dimension $n$ . A chart $(U, \varphi)$ on $M$ is an open set $U \subseteq M$ with a homeomorphism $\varphi$ from $U$ onto an open subset of $\R^n$ . Two charts $(U, \varphi)$ and $(V, \psi)$ are smoothly compatible if $U \cap V = \emptyset$ , or if the transition map $\psi \circ \varphi^{-1} \colon \varphi(U \cap V) \to \psi(U \cap V)$ is smooth with smooth inverse. A smooth atlas on $M$ is a family of pairwise smoothly compatible charts whose domains cover $M$ . A smooth structure on $M$ is a smooth atlas $\mathcal{A}$ that is maximal. This means that every chart that is smoothly compatible with all charts of $\mathcal{A}$ already belongs to $\mathcal{A}$ .

Remarks

Every smooth atlas is contained in a unique maximal smooth atlas, so a smooth structure is usually given by any one atlas. Example: on $\R^n$ the single chart $(\R^n, \id)$ is a smooth atlas, and the maximal atlas containing it is the standard smooth structure. Non-example: the atlas $\set{(\R, \id), (\R, t \mapsto t^3)}$ on $\R$ is not a smooth atlas. The transition map $t \mapsto t^{1/3}$ is not differentiable at $0$ , so the two charts are not smoothly compatible. Each of the two charts alone defines a smooth structure; the two structures are different but diffeomorphic. Two smooth structures on $M$ are usually counted as the same when they are diffeomorphic, not only when they are equal.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Atlas concept
Homeomorphism concept
Smooth function concept
Topological manifold concept
Smooth structure definition

Tag	`VG4ZVTGJ9W`
URI	`m2://VG4ZVTGJ9W`
Kind	definition
Writing	W0 stub
MSC	57R55, 58A05
Defines	Smooth structure
Uses	Topological manifold, Atlas, Homeomorphism, Smooth function
Changed	2 November 2026

Smooth structure

Definition

Remarks

Reading path

Further reading

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