2024/04/01 by Lee-Peng Teo, Teo, Lee-Peng
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #History and Overview (math.HO) #Mathematics and Applications #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2404.00835
openalex publication_date 2024/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.