Smooth atlas

Note: a smooth atlas is a special kind of Atlas

Definition

An atlas $\mathcal{A}$ is called a smooth atlas if:

• Any two charts in $\mathcal{A}$ are smoothly compatible with each other.

Maximal

A smooth atlas $\mathcal{A}$ on $M$ is maximal if it is not properly contained in any larger smooth atlas. This means every smoothly compatible chart with a chart in $\mathcal{A}$ is already in $\mathcal{A}$

Complete

A complete smooth atlas is a synonym for maximal smooth atlas

We can now define a Smooth manifold

Verifying an atlas is smooth

First way

You need only show that that each Transition map is Smooth for any two charts in $\mathcal{A}$, once this is done it follows the transition maps are diffeomorphisms because the inverse is already a transition map.

Second way

Given two particular charts $(U,\varphi)$ and $(V,\psi)$ is may be easier to show that they are smoothly compatible by verifying that $\psi\circ\varphi^{-1}$ is smooth and injective with non-singular Jacobian at each point. We can then use

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TODO: C.36 - Introduction to smooth manifolds - second edition

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See also

• Motivation for smooth structures
• Smooth
• Diffeomorphism
• Transition map
• Smoothly compatible charts
• Topological manifold
• Smooth manifold