Manifold and maximal atlas 1) I didn't understand really what is a maximal atlas. Is it as set of compatible chart maximal in the sens that adding one more chart will yield the atlas not compatible ? 2) Let two atlas $\mathcal A$ and $\mathcal A'$. So if they are compatible, they are both in a maximal atlas $\hat{\mathcal A}$ ? 3) And if they are not compatible, there are two atlas $\hat{\mathcal A}$ and $\tilde{\mathcal A}$ such that $\mathcal A$ is for example in $\hat{\mathcal A}$ and $\mathcal A'\in\tilde{\mathcal A}$ ? 4) And if I understood well, $\hat{\mathcal A}$ gives smooth structure and $\tilde{\mathcal A}$ gives an other smooth structure ? But both are incompatible ? I hope my question are clear enough.

(1) Yes, by the definition of maximal.

(2) Yes. Technical detains in Is Zorn's lemma required to prove the existence of a maximal atlas on a manifold? and Why maximal atlas.

(3) I understand that the "they" in "And if they are not compatible..." are two _charts_. Yes, each chart is in an atlas _and the intersection of both atlas is empty_.

(4) They give _different_ structures. But can be _diffeomorphic_. Easy example: $\Bbb R$ and the two atlases $\\{x\longmapsto x\\}$ and $\\{x\longmapsto x^3\\}$.