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Mat. Sb., 1990, Volume 181, Number 12, Pages 1710–1720 (Mi msb1256)  

On the nonbendability of closed surfaces of trigonometric type

Yu. A. Aminov

Physical Engineering Institute of Low Temperatures, UkrSSR Academy of Sciences

Abstract: 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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English version:
Mathematics of the USSR-Sbornik, 1992, 71:2, 549–560

Bibliographic databases:

UDC: 514
MSC: 53A05
Received: 08.12.1988

Citation: 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
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\paper On the nonbendability of closed surfaces of trigonometric type
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\vol 181
\issue 12
\pages 1710--1720
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..71..549A}
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\jour Math. USSR-Sb.
\yr 1992
\vol 71
\issue 2
\pages 549--560
\crossref{https://doi.org/10.1070/SM1992v071n02ABEH001408}
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