Abstract
We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.
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References
O Bonnet. Mémoire sur la theorie des surfaces applicables sur une surface donnée. J l’ École Pol, Paris (1867), XLII Cahier, 72–92
É Cartan. Sur la couples de surfaces applicables avec conservation des courbures principales. Bull Soc Math, 1942, 66: 55–85, or Oeuvres Complètes, III(2), 1591–1620
S S Chern. Deformation of surfaces preserving principal curvatures. Differential Geometry and Complex Analysis. H E Rauch Memorial Volume, Springer Verlag, 1985, 155–163
H B Lawson Jr. Complete minimal surfaces in S3. Ann of Math Math, 1970, 92: 335–374
K Voss. Bonnet Surfaces in Spaces of Constant Curvature. Japan: Lecture Notes II of 1st MSJ Research Institute, Sendai, 1993, 295–307
A I Bobenko, U Eitner. Bonnet surfaces and Painlevé Equations. SFB 288 Preprint 151, TU Berlin, 1994
X Chen, C K Peng. Deformation of surfaces preserving principal curvatures. Lecture Notes in Math, 1369, Springer Verlag (Differential Geometry and Topology Proceedings, Tianjin, 1986–1987), 63–70
K Kenmotsu. An intrinsic characterization ofH-deformable surfaces. J London Math Soc, 1994, 49(2): 555–568
I M Roussos. The helicoidal surfaces as Bonnet surfaces. Tôhoku Math J, Second Series, 1988, 40(3): 486–490
I M Roussos. Global results on Bonnet surfaces. Journal of Geometry, 1999
Z Soyuçok. The problem of non-trivial isometries of surfaces preserving the principal curvatures. J of Geometry, 1995, 52: 173–188
G Kamberov, F Pedit, U Pinkall. Bonnet pairs and isothermic surfaces. Preprint
H B Lawson Jr, R de A Tribuzy. On the mean curvature function for compact surfaces. J Differential Geometry, 1981, 16: 179–183
S H Doong, I M Roussos. Isometric immersions of tori intoR 3 with given mean curvature. Results in Mathematics (Resultate der Mathematik), 1992, 21(3/4): 313–318
I M Roussos. A problem of rigidity for compact surfaces. Proceedings of the 4th International Congress of Geometry, Academy of Athens, Aristotle University of Thessaloniki, Thessaloniki 1996, 366–372
I M Roussos. Principal curvature preserving isometries of surfaces in ordinary space. Bol Soc Bras Mat, 1987, 18(2): 95–105
L P Eisenhart. A Treatise on the Differential Geometry of Curves and Surfaces. Boston, USA: Ginn and Company, 1909
I M Roussos, G E Hernandez. On the number of distinct isometric immensions of a Riemannian Surface in R3 with given mean curvature. Am J Math, 1990, 112: 71–85
R de A Tribuzy. A characterization of tori with constant mean curvature in a space form. Bol Soc Bras Math, 1980, II: 259–274
M Umehara. A characterization of compact surfaces with constant mean curvature. Proc AMS, 1990, 108: 483–489, 1993, 295–307
A G Colares, K Kenmotsu. Isometric deformation of surfaces in R3 preserving the mean curvature function. Pacific J Math, 1989, 136(1): 71–80
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Dedication Dedicated to Siuping Ho for all her invaluable support and encouragement.
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Roussos, I.M. Tangential developable surfaces as bonnet surfaces. Acta Mathematica Sinica 15, 269–276 (1999). https://doi.org/10.1007/BF02650670
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DOI: https://doi.org/10.1007/BF02650670