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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characteristic property of the sphere


Author: Giacomo Saban
Journal: Proc. Amer. Math. Soc. 86 (1982), 123-125
MSC: Primary 53A05; Secondary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1982-0663880-X
MathSciNet review: 663880
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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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DOI: https://doi.org/10.1090/S0002-9939-1982-0663880-X
Article copyright: © Copyright 1982 American Mathematical Society