In particular, a smooth structure allows mathematical analysis to be performed on the manifold.
This gives a natural equivalence relation on the set of smooth atlases.
By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas.
This atlas contains every chart that is compatible with the smooth structure.
In general, computations with the maximal atlas of a manifold are rather unwieldy.
For most applications, it suffices to choose a smaller atlas.
For example, if the manifold is compact, then one can find an atlas with only finitely many charts.
[citation needed] John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure.
This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.
-times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic.
or (real-)analytic structure on the manifold rather than a smooth one.
Similarly, a complex structure can be defined by requiring the transition maps to be holomorphic.