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