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I some how could not find the definition of maximal atlas on a manifold.

What I see is that an atlas is said to be maximal atlas if it is not contained in any other atlas.

What does this containment actually mean?

Let $\mathcal{A}$ be an atlas and $\mathcal{B}$ be another atlas. When do we say that $\mathcal{A}$ is contained in $\mathcal{B}$?

I was not able to find definition of this.

Another confusion is about union of atlases. Let $\mathcal{A}$ and $\mathcal{B}$ be two atlases. What do we mean by union of atlases? Is it just the union $\{(U,\phi)_{\phi\in \mathcal{A}},(V,\psi)_{\psi\in \mathcal{B}}\}$?

It may happen that this union is not an atlas i.e., there can be two charts $\phi_\mathcal{A}$ and $\psi_{\mathcal{B}}$such that $\phi_{\mathcal{A}}$ and $\psi_{\mathcal{B}}$ are not compatible.

By maximal atlas do I mean an atlas $\mathcal{A}$ such that for any other atlas $\mathcal{B}$, the union as above is not an atlas?

Any reference for the definition is welcome.

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    Can you say what your definition of "atlas" is? Your notation ${U_p,\phi_p}_{p\in M}$ is unusual; by the definition I know an atlas is not indexed by the elements of $M$... – Eric Wofsey Feb 27 '17 at 05:17
  • @EricWofsey : I have edited the question. An atlas is not indexed by elements of $M$. An atlas is a collection of charts $(U,\phi)$ such that union of these open sets $U$ is $M$ and given any two charts $(U,\phi)$ and $(V,\psi)$ the transition maps are smooth. –  Feb 27 '17 at 05:22

2 Answers2

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"Contain" and "union" here literally mean just that. An atlas $\mathcal{A}$ is a set of charts $(U,\phi)$, and $\mathcal{A}$ is contained in $\mathcal{B}$ if $\mathcal{A}\subseteq\mathcal{B}$: that is, if every chart which is an element of $\mathcal{A}$ is also an element of $\mathcal{B}$. The union of two atlases is just the set $\mathcal{A}\cup\mathcal{B}$, which as you observe may not be an atlas.

An atlas $\mathcal{A}$ is called maximal if there does not exist any atlas $\mathcal{B}$ such that $\mathcal{A}\subset\mathcal{B}$ (with a strict inclusion). This is equivalent to saying that if $\mathcal{B}$ is an atlas such that $\mathcal{A}\cup\mathcal{B}$ is an atlas, then $\mathcal{B}\subseteq\mathcal{A}$.

Eric Wofsey
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Let $\mathcal{A} \\$ be an atlas, we will say that $\mathcal{A} \\$ is a maximal atlas if for every pair ($ { U,ϕ } $) where $ϕ(U)$, $ψ(U\cap V)$ and $ϕ(U\cap V)$ are open subsets with the usual topology for every chart ($ { V,ψ } $) of $\mathcal{A} \\$ then the transition maps are diffeomorfism

Pablo Borrego
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