# How to show an atlas is maximal

by Identity
Tags: atlas, maximal
 P: 150 I've been looking in various books in differential geometry, and usually when they show that a smooth manifold has a differentiable structure, they just show that the atlas is $C^\infty$ compatible, and forget about showing it is maximal. Which got me thinking. Given an atlas, how DOES one show that it is maximal? After all, you need to show that a completely arbitrary chart that is not in the atlas cannot be compatible with the atlas. But how do you show compatibility if you don't even know what this chart is? And isn't this an important step in showing that a manifold has a differentiable structure?
 Sci Advisor HW Helper P: 9,499 i don't know the answer, but i cannot think of any situation where i have ever cared about the answer to this question. i.e. all you need is an atlas that covers the manifold. then you can study the manifold. of course every question is interesting, so maybe i can think more about it. but actually all you want is any compatible atlas that covers the manifold. then you just enlarge it to a maximal one if you wish. i.e. a differentiable structure exists if and only if an atlas exists that covers. then a maximal one also exists.
 P: 150 Thanks, yeah I also thought that having a maximal atlas wouldn't be that important as long as you had compatible charts. But anyway every text I read has it as a condition, which I thought was interesting: "A (smooth) differential manifold $M^m$ of dimension m is a topological manifold of dimension m together with a maximal (smooth) atlas on it." - An Introduction to Differential Manifolds, Barden, Thomas "A maximal $C^\infty$ atlas $A$ on $M$ is called a smooth structure on $M$" - Differentiable Manifolds, Conlon "A smooth or $C^\infty$ manifold is a topological manifold $M$ together with a maximal atlas." - An Introduction to Manifolds, Tu A maximal atlas is also the third condition for a differentiable structure (after covering and compatible charts) in my lecture notes.
 P: 161 How to show an atlas is maximal assume that it isn't maximal. so you have another mapping on an object with differentiable structure. but wait! you didn't add it to the collection! therefore maximal.