The Brioschi Formula for the Gaussian Curvature

Lee-Peng Teo

Published 2024-04-01, updated 2024-04-02Version 2

The Brioschi formula expresses the Gaussian curvature $K$ in terms of the functions $E, F$ and $G$ in local coordinates of a surface $S$. This implies the Gauss' theorema egregium, which says that the Gaussian curvature just depends on angles, distances, and their rates of change. In most of the textbooks, the Gauss' theorema egregium was proved as a corollary to the derivation of the Gauss equations, a set of equations expressing $EK, FK$ and $GK$ in terms of the Christoffel symbols. The Christoffel symbols can be expressed in terms of $E$, $F$ and $G$. In principle, one can derive the Brioschi formula from the Gauss equations after some tedious calculations. In this note, we give a direct elementary proof of the Brioschi formula without using Christoffel symbols. The key to the proof are properties of matrices and determinants.