The metric \(\mathring{g}=\iota^*\bar{g}\) induced on \(\mathbb{S}^n\) by the inclusion \(\iota\colon \mathbb{S}^n \hookrightarrow \mathbb{R}^{n+1}\) is called round metric.
Remark
- the round metric on \(\mathbb{S}^2\) is not flat [@lee2013smooth_manifolds, Corollary 13.20]