A Characteristic Property of the Sphere

Abstract

Let $S$ be any connected piece of surface in Euclidean three-space, of class ${C^3}$ and ${g_{ij}}$, ${l_{ij}}$ be the coefficients of the first and second fundamental forms of $S$. If these coefficients satisfy the system of differential equations obtained by interchanging the ${g_{ij}}$ and ${l_{ij}}$ having same indices in the Mainardi-Codazzi equations, $S$ is part of a sphere. Furthermore, if two metrics on $S$ satisfy a similar condition, they are proportional.