Difference between revisions of "Smooth atlas"
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Latest revision as of 12:27, 12 November 2015
Note: a smooth atlas is a special kind of Atlas
Contents
Definition
An atlas [ilmath]\mathcal{A} [/ilmath] is called a smooth atlas[1] 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
TODO: C.36 - Introduction to smooth manifolds - second edition
See also
- Motivation for smooth structures
- Smooth
- Diffeomorphism
- Transition map
- Smoothly compatible charts
- Topological manifold
- Smooth manifold
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
