Smooth Manifold

A topological manifold equipped with an atlas whose transition maps are smooth.

Smooth Manifold

A smooth manifold is a topological space that locally looks like Euclidean space and whose coordinate changes are differentiable to all orders.

Setup

Let $M$ be a topological space . A chart of dimension $n$ is a pair $(U,\varphi)$ where $U \subseteq M$ is an open set and $\varphi:U\to \varphi(U)\subseteq \mathbb{R}^n$ is a homeomorphism.

An atlas is a collection of charts $\{(U_\alpha,\varphi_\alpha)\}$ whose domains cover $M$ .

Smoothness condition

An atlas is smooth (or $C^\infty$ ) if for any overlapping charts the transition map

\varphi_\beta\circ \varphi_\alpha^{-1}:\ \varphi_\alpha(U_\alpha\cap U_\beta)\to \varphi_\beta(U_\alpha\cap U_\beta)

is a smooth map between open subsets of $\mathbb{R}^n$ (in the usual multivariable calculus sense).

A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold (typically assumed Hausdorff and second countable; see Hausdorff ) and $\mathcal{A}$ is a smooth atlas. Two atlases are considered equivalent if their union is still smooth; each equivalence class has a unique maximal smooth atlas.

Key properties

The integer $n$ is the dimension of $M$ , written $\dim M=n$ .
A smooth structure lets you define smooth functions $f:M\to \mathbb{R}$ , smooth maps between manifolds, and the tangent space at each point.

Examples

$\mathbb{R}^n$ with the single global chart $(\mathbb{R}^n,\mathrm{id})$ .
Any open subset $U\subseteq \mathbb{R}^n$ (with the inherited topology) is a smooth manifold.
The sphere $S^n$ admits smooth atlases (e.g., via stereographic projection).