On the nonbendability of closed surfaces of trigonometric type
Yu. A. Aminov
Physical Engineering Institute of Low Temperatures, UkrSSR Academy of Sciences
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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Mathematics of the USSR-Sbornik, 1992, 71:2, 549–560
Yu. A. Aminov, “On the nonbendability of closed surfaces of trigonometric type”, Mat. Sb., 181:12 (1990), 1710–1720; Math. USSR-Sb., 71:2 (1992), 549–560
Citation in format AMSBIB
\paper On the nonbendability of closed surfaces of trigonometric type
\jour Mat. Sb.
\jour Math. USSR-Sb.
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