A characteristic property of the sphere

Saban, Giacomo

doi:10.1090/S0002-9939-1982-0663880-X

A characteristic property of the sphere
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Proc. Amer. Math. Soc. 86 (1982), 123-125 Request permission

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.

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