Geodesic
On the nonbendability of closed surfaces of trigonometric type
Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 549-560

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In connection with a well-known problem on the existence of closed bendable surfaces in $E^3$ the author considers the class of surfaces for which each component of the radius vector is a trigonometric polynomial in two variables. Two theorems on the nonbendability of surfaces in this class are proved, and an expression for the volume of the domain bounded by such a surface is established. Theorem 1 (the main theorem) asserts the nonbendability of a surface under the condition that some Diophantine equation does not have negative solutions. In this case the coefficients of the second fundamental form can be expressed in a finite-valued way in terms of the coefficients of the first fundamental form as algebraic expressions. 
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     author = {Yu. A. Aminov},
     title = {On the nonbendability of closed surfaces of trigonometric type},
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Yu. A. Aminov. On the nonbendability of closed surfaces of trigonometric type. Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 549-560. http://geodesic.mathdoc.fr/item/SM_1992_71_2_a17/