A unit first-fundamental-form determinant means area is preserved, not necessarily angles

Let U be an open subset of $\mathbb{R}^2$ and S be a smooth surface in $\mathbb{R}^3$. If $\sigma:U\to S$ is a regular surface patch such that $\det\left(\begin{smallmatrix}\sigma_u\cdot\sigma_u&\sigma_u\cdot\sigma_v
\sigma_u\cdot\sigma_v&\sigma_v\cdot\sigma_v\end{smallmatrix}\right)=1$. Then: