Let \(M_n\) be a \(n\)-dimensional manifold . An atlas \((\Omega_i, \varphi_i)_{i \in I}\) is of class \(C^k\), for \(k\in \mathbb{N}\), \(C^\infty\) or \(C^\omega\) if every transition map \(\psi\circ \phi^{-1}\) with nonempty domain is a \(C^k\)-, \(C^\infty \)-, or a \(C^\omega\)-diffeomorphism, respectively. In this case, we call \(\psi\) and \(\phi\) smoothly compatible.
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.
- Every atlas containing exactly on chart is a smooth one by definition.
Example