Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 549-560
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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/
@article{SM_1992_71_2_a17,
author = {Yu. A. Aminov},
title = {On the nonbendability of closed surfaces of trigonometric type},
journal = {Sbornik. Mathematics},
pages = {549--560},
year = {1992},
volume = {71},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1992_71_2_a17/}
}
TY - JOUR
AU - Yu. A. Aminov
TI - On the nonbendability of closed surfaces of trigonometric type
JO - Sbornik. Mathematics
PY - 1992
SP - 549
EP - 560
VL - 71
IS - 2
UR - http://geodesic.mathdoc.fr/item/SM_1992_71_2_a17/
LA - en
ID - SM_1992_71_2_a17
ER -
%0 Journal Article
%A Yu. A. Aminov
%T On the nonbendability of closed surfaces of trigonometric type
%J Sbornik. Mathematics
%D 1992
%P 549-560
%V 71
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1992_71_2_a17/
%G en
%F SM_1992_71_2_a17
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.
