Together with the standard smooth structure on \(\mathbb{R}^n\) , \(\mathbb{R}^n\) is a smooth manifold .
Remarks
- The flat space \(\mathbb{R}^n\), for \(n\neq 4\), has only the standard smooth structure for every \(n\neq 4\). On \(\mathbb{R}^4\) there exists uncountably many smooth structures which are not diffeomorphic to each other (Donaldson-Freedman, 1984). The Euclidean space with nonstandard structures are called fake \(\mathbb{R}^4\)’s.