Is valid the Brioschi Formula of Gaussian curvature in $R^n$?

Question

Is the Brioschi formula, for Gaussian Curvature of a Surface, valid only in codimension 1, or is valid for a surface in $R^n$ with $n>3$?

Answer 1

Yes, the Brioschi formula for $K$ in terms of the first fundamental form is valid for any codimension. The point is that $K$ depends only on the induced metric, not on the second fundamental form, so it is completely specified by knowing the functions $E$, $F$, and $G$ in a given local $uv$-coordinate system. When $E = x_u\cdot x_u$, $F=x_v\cdot x_v$, and $G = x_v\cdot x_v$ are given by $x:S\to \mathbb{R}^n$, the ambient dimension $n$ doesn't matter.