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 [itex]C^\infty[/itex] 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?