# On the problem of isometry of a hypersurface preserving mean curvature

Article

First Online:

## Abstract

The problem of determining the *Bonnet hypersurfaces in R* ^{ n+1}, for *n* > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called *Bonnet associate* of the initial hypersurface.

The orthogonal net which is called *A-net* is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in *R* ^{ n+1} with no umbilical points is a Bonnet hypersurface if and only if it has an *A*-net.

## Keywords

Bonnet hypersurface Bonnet associate isometry mean curvature preserving Bonnet curve*A*-net

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bobenko A I and Either U, Bonnet surfaces and Painlevé equations,
*J. Reine Angew. Math.***499**(1998) 47–79zbMATHMathSciNetGoogle Scholar - [2]Bonnet O, Memoire sur la theorie des surfaces, Applicable sur une surface donne,
*J. Ec. Poyt.*(1867) T. 25Google Scholar - [3]Cartan E, Sur les couples surfaces applicables avec conservation des courbures principales,
*Bull. Sci. Math.***66**(1942) 55–163MathSciNetGoogle Scholar - [4]Chern S S, Deformations of surfaces preserving principal curvature, Differential Geometry and Complex Analysis, H. E. Rauch Memorial Volume (eds) I Chavel and H M Farkas (1985) (Springer-Verlag) pp. 155–163Google Scholar
- [5]Chern S S, Bryant R L, Gardner R B, Goldschmidt H L and Griffiths P A, Exterior Differential Systems, Mathematical Sciences Research Institute Publications (1991) (New York: Springer-Verlag)Google Scholar
- [6]Colares A G and Kenmotsu K, Isometric deformation of surfaces in preserving the mean curvature function,
*Pacific J. Math.***136**(1989) 71–80zbMATHMathSciNetGoogle Scholar - [7]Csikos B, Differential Geometry, Lectures Notes, Budapest Semesters in Mathematics (1998)Google Scholar
- [8]Kenmotsu K, An intrinsic characterization of
*H*-deformable surfaces,*J. London Math. Soc.***49**(1994) 555–568zbMATHMathSciNetGoogle Scholar - [9]Kobayashi S and Nomizu K, Foundations of differential geometry (1969) (New York: Interscience) vol. 2zbMATHGoogle Scholar
- [10]Kokubu M, Isometric deformations of hypersurfaces in a Euclidean space preserving mean curvature,
*Tohoku Math. J.***44**(1992) 433–442zbMATHMathSciNetGoogle Scholar - [11]Roussos I M, Principal curvature preserving isometries of surfaces in ordinary spaces,
*Bol. Soc. Math.***18(2)**(1987) 95–105zbMATHCrossRefMathSciNetGoogle Scholar - [12]Roussos I M, Global results on Bonnet surfaces,
*J. Geom.***65**(1999) 151–158zbMATHCrossRefMathSciNetGoogle Scholar - [13]Soyuçok Z, The problem of non-trivial isometries of surfaces preserving principal curvatures,
*J. Geom.***52**(1995) 173–188zbMATHCrossRefMathSciNetGoogle Scholar - [14]Soyuçok Z, The problem of isometric deformations of a Euclidean hypersurface preserving mean curvature,
*Bull. Tech. Univ.***49**(1996) 551–562zbMATHGoogle Scholar - [15]Voss K, Bonnet surfaces in spaces of constant curvature, Lecture Notes II of 1st MSJ Research Instıtute, Sendai, Japan (1993) pp. 295–307Google Scholar
- [16]Xiuxiong C and Chia-Kuei P, Deformation of surfaces preserving principal curvatures,
*Lect. Notes Math.*(1989) 63–70Google Scholar

## Copyright information

© Indian Academy of Sciences 2007