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 Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 163–176 (Mi msb3646)

Canonical $A$-deformations preserving the lengths of lines of curvature on a surface

L. L. Beskorovainaya

Abstract: In this paper, infinitesimal deformations which preserve the area element of a surface in $E_3$ ($A$-deformations) which also preserve the lengths of lines of curvature are studied. Here $A$-deformations are considered up to infinitesimal bendings (which constitute the trivial case for the problem posed). Such $A$-deformations are also called canonical.
For regular surfaces of nonzero total curvature (without umbilic points) the problem indicated reduces to a homogeneous second order partial differential equation of elliptic type. In this paper a series of results about the existence and arbitrariness of canonical $A$-deformations is obtained. The basic results are valid for surfaces in the large.
Bibliography: 20 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 151–164

Bibliographic databases:

UDC: 513.013
MSC: Primary 53A05; Secondary 35J25, 73L99

Citation: L. L. Beskorovainaya, “Canonical $A$-deformations preserving the lengths of lines of curvature on a surface”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 163–176; Math. USSR-Sb., 26:2 (1975), 151–164

Citation in format AMSBIB
\Bibitem{Bes75} \by L.~L.~Beskorovainaya \paper Canonical $A$-deformations preserving the lengths of lines of curvature on a~surface \jour Mat. Sb. (N.S.) \yr 1975 \vol 97(139) \issue 2(6) \pages 163--176 \mathnet{http://mi.mathnet.ru/msb3646} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=394456} \zmath{https://zbmath.org/?q=an:0323.53001} \transl \jour Math. USSR-Sb. \yr 1975 \vol 26 \issue 2 \pages 151--164 \crossref{https://doi.org/10.1070/SM1975v026n02ABEH002474} 

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This publication is cited in the following articles:
1. V. T. Fomenko, “A property of conformal infinitesimal deformations of multidimensional surfaces in Riemannian space”, Math. Notes, 59:2 (1996), 201–204
2. Mikes J. Stepanova E. Vanzurova A., “Differential Geometry of Special Mappings”, Differential Geometry of Special Mappings, Palacky Univ, 2015, 1–566
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