Abstract
In this paper we deal with the Bonnet problem of determining the surfaces in the Euclidean three dimensional space which can admit at least one nontrival isometry that preserves the principal curvatures(Bonnet surfaces). The problem is considered locally and examined in the general case. The main results are: (a) Necessary and sufficient condition for a surface to be a Bonnet surface is that it admits a special isothermal parameter system. (b) Complete solution of the problem in the class of the isothermic surfaces. Moreover: These results and the methods used provide a new efficient and elegant manner of proving the, already known, fact that all helicoidal surfaces are Bonnet surfaces and determine the already known developable Bonnet surfaces.
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References
BONNET, O.: Memoire sur la theorie des surfaces, Applicable sur une surface donnee, Journale de l'Ecole Politechnique, T.25, 1867.
GRAUSTEIN, W. C.: Applicability with preservation of both curvatures, Bull. Amer. Math. Soc., 30, 19–23, 1924.
CARTAN, E.: Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. Sci. Math. 66, 55–85, 1942.
TRIBUZY, R.: A characterization of tori with constant mean curvature in a space form, Bol. Soc. Brasil. Mat. 11, 259–274, 1980.
LAWSON, H. Bl.and TRIBUZY, R.: On the mean curvature function for compact surfaces, J. Diff. Geometry, 16, 179–183, 1981.
CHERN, S. S.: Deformation of surfaces preserving principal curvatures, Differantial geometry and comlexX analysis, 155–163, 1985.
ROUSSOS, I. M.: Mean curvature preserving isometries of surfaces in ordinary space, Ph. D. Thesis, University of Minnesota, March 1986.
ROUSSOS, I. M.: Principal curvature preserving isometries of surfaces in ordinary space, Bol. Soc. Mat. 18,2,95–105, 1987.
ROUSSOS, I. M.: The helicoidal surfaces as Bonnet surfaces, Tohoku Math. J., 40, 485–490, 1988.
XIUXIONG, C.and CHIA-KUEI, P.: Deformation of surfaces preserving principal curvatures, Lect. Notes Math., 63–70, 1989.
COLARES, A. G.and KEMNOTSU, K.: Isometric deformation of surfaces inR 3 preserving the mean curvature function, Pasific J.Math.,136, 71–80, 1989.
UMEHARA, M.: A characterization of compact surfaces withconstant mean curvature, Proc. AMS, 108, 483–489, 1990.
ROUSSOS, I. M.and HERNANDEZ, G. E.: On the number of distinct isometric immersions of a Riemannian surface intoR 3 with given mean curvature, Am.J.Math., 112, 71–85, 1990.
EISENHART, L. P.: A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, Inc.,New York, 1960.
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Soyuçok, Z. The problem of non-trivial isometries of surfaces preserving principal curvatures. J Geom 52, 173–188 (1995). https://doi.org/10.1007/BF01406838
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DOI: https://doi.org/10.1007/BF01406838