The euclidean metric is defined by the euclidean distance, i.e. \(d(x,y)=\lvert x-y\rvert\).
The euclidean metric on \(\mathbb{R}^n\) in the Riemannian sense is defined by
\begin{equation*} g=\delta_{ij}dx^i\otimes dx^j. \end{equation*}Then for \(v,w\in \mathbb{R}^n\) we obtain
\begin{equation*} g(v,w)=\delta_{ij}v^iw^j=\sum_{i=1}^{n} v^iw^i=\langle v, w\rangle, \end{equation*}i.e. the euclidean metric coincide with the usual scalar product.
Notation: \(\bar{g}\)