Geodesic
Isometric surfaces with a~common mean curvature and the problem of Bonnet pairs
Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 111-152

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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. 
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I. Kh. Sabitov. Isometric surfaces with a~common mean curvature and the problem of Bonnet pairs. Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 111-152. http://geodesic.mathdoc.fr/item/SM_2012_203_1_a5/