Smooth Manifolds

Smooth Atlases

Definition: Smooth Atlas

An atlas is smooth iff all of its charts are $C^{\infty }$-compatible.

Smooth Structures

Definition: Smooth Structure

A smooth structure on a manifold is a maximal smooth atlas.

NOTE

Smooth structures are also known as differentiable structures.

Smooth Manifolds

Definition: Smooth Manifold

A smooth manifold $(M,\mathcal{S})$ is a manifold $M$ equipped with a smooth structure $\mathcal{S}$.

NOTE

Although a smooth manifold technically has two components - a manifold $M$ and a smooth structure $\mathcal{S}$, - if $\mathcal{S}$ is understood from context we usually this technicality and just call $M$ a smooth manifold.

NOTE

Smooth manifolds are also known as differentiable manifolds.

Smooth Charts

Definition: Smooth Chart

Let $(M,\mathcal{S})$ be a smooth manifold.

A smooth chart is any chart $(U,\varphi )$ contained in the smooth structure $\mathcal{S}$.

Definition: Smooth Coordinate System

The coordinate system of a smooth chart is known as a smooth coordinate system.