Let \(\mathbb{S}^n\) denote the \(n\)-sphere . Together with the standard smooth structure on \(\mathbb{S}^n\) , \(\mathbb{S}^n\) is a [smooth manifold](smooth manifold.md).
Remarks
- Alternatively, \(\mathbb{S}^n\) is smooth since it is a regular level set .
- In fact, the sphere is even a real analytic manifold with the standard analytic structure .
- For \(n\le 6\) and \(n\neq 4\), \(\mathbb{S}^n\) has only the standard smooth structure up to diffeomorphism. For \(n=7\) there are 15 (Milnor, 1956).