Smooth atlas

Note: a smooth atlas is a special kind of Atlas

Definition

An atlas [ilmath]\mathcal{A} [/ilmath] is called a smooth atlas if:

• Any two charts in [ilmath]\mathcal{A} [/ilmath] are smoothly compatible with each other.

Maximal

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

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 [ilmath]\mathcal{A} [/ilmath], 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 [ilmath](U,\varphi)[/ilmath] and [ilmath](V,\psi)[/ilmath] is may be easier to show that they are smoothly compatible by verifying that [ilmath]\psi\circ\varphi^{-1} [/ilmath] 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