A smooth atlas ๐ on a manifold \( M \) is maximal or complete if it is not properly contained in any larger smooth atlas. In particular, if a chart \( (\varphi, U) \) is compatible with every chart in ๐, then \( (\varphi, U) \in ๐ \). A smooth structure on \( M \) is a maximal smooth atlas on \(M\).
Accordingly, \(C^k\)-, \(C^{k,\alpha}\)- and analytic structures are defined.
Remarks
- Every smooth atlas is contained in a unique maximal one.
- Two smooth atlases determine the same smooth structure if and only if the union is a smooth atlas.
- Another way to determine a smooth structure on \(M\) is to define an equivalence relation and to use the corresponding equivalence classes as a smooth structure on \(M\).