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 Mat. Sb., 2012, Volume 203, Number 1, Pages 115–158 (Mi msb7706)

Isometric surfaces with a common mean curvature and the problem of Bonnet pairs

I. Kh. Sabitov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Simple methods are used to give new proofs, and sometimes to make them more precise, of basic theorems on isometric surfaces with a common mean curvature, which are usually called Bonnet pairs. The considerations are conducted under the assumption of minimally admissible smoothness of the objects in question, and certain necessary or sufficient criteria are given for the non-existence of Bonnet pairs with a common non-constant mean curvature among compact surfaces.
Bibliography: 26 titles.

Keywords: surfaces, isometry, mean curvature, invariance.

DOI: https://doi.org/10.4213/sm7706

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English version:
Sbornik: Mathematics, 2012, 203:1, 111–152

Bibliographic databases:

UDC: 514.772.35
MSC: Primary 53A05; Secondary 53A10, 53C42
Received: 09.03.2010 and 09.04.2011

Citation: I. Kh. Sabitov, “Isometric surfaces with a common mean curvature and the problem of Bonnet pairs”, Mat. Sb., 203:1 (2012), 115–158; Sb. Math., 203:1 (2012), 111–152

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7706
• https://doi.org/10.4213/sm7706
• http://mi.mathnet.ru/eng/msb/v203/i1/p115

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. A. Zagryadskii, “The relations between the Bertrand, Bonnet, and Tannery classes”, Moscow University Mathematics Bulletin, 69:6 (2014), 277–279
2. M. T. Anderson, “Static vacuum Einstein metrics on bounded domains”, Ann. Henri Poincaré, 16:10 (2015), 2265–2302
3. M. T. Anderson, “Conformal immersions of prescribed mean curvature in $\mathbb R^3$”, Nonlinear Anal., 114 (2015), 142–157
4. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175
5. N. Ando, “Over-determined systems in relation to principal curvatures”, J. Geom., 108:2 (2017), 355–373
6. Jensen G.R., Musso E., Nicolodi L., “Compact Surfaces With No Bonnet Mate”, J. Geom. Anal., 28:3 (2018), 2644–2652
7. He H.X., Ma H., Wang E.X., “Lagrangian Bonnet Problems in Complex Space Forms”, Acta. Math. Sin.-English Ser., 35:8 (2019), 1357–1366
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