Timelike Bonnet surfaces with non-constant curvature

Year 2013, Volume: 17 Issue: 1, 113 - 118, 01.04.2013

Soley Ersoy Kemal Eren

Abstract

In this study, the criterion of a timelike surface being Bonnet surface in 3-dimensional Minkowski space given by [11] is taken into consideration and by a similar manner of the classification of surfaces in Euclidean space done by I. M. Roussos in [6], timelike surfaces as Bonnet surfaces are investigated in three class as C , C and C . Timelike surfaces given in the case of C have constant mean curvature and were investigated by a detailed way in [11]. In the present study, by investigating the cases of C and C , a criterion of the timelike surfaces with non-constant mean curvature being Bonnet surfaces is determined.

Keywords

Timelike Bonnet Surfaces, mean curvature, Gaussian curvature, isometry

Sabit olmayan ortalama eğrilikli timelike Bonnet yüzeyler

Abstract

In this study, the criterion of a timelike surface being Bonnet surface in 3-dimensional Minkowski space given by [11] is taken into consideration and by a similar manner of the classification of surfaces in Euclidean space done by I. M. Roussos in [6], timelike surfaces as Bonnet surfaces are investigated in three class as C1, C2 and C3. Timelike surfaces given in the case of C1 have constant mean curvature and were investigated by a detailed way in [11]. In the present study, by investigating the cases of C2 and C3, a criterion of the timelike surfaces with non-constant mean curvature being Bonnet surfaces is determined.

Keywords

Timelike Bonnet yüzeyler, ortalama eğrilik, Gauss eğriliği, izometri

References

• BONNET, O., Mémoire sur la théorie des surfaces applicables, J. École Polytech. 42 (1867), 72-92.
• CARTAN, E., Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. Sc. Math. 66 (1942), 1-30; reprinted in: Oeuvres Completes, Partie III, vol.2, 1591-1620.
• LAWSON, H. B., Complete minimal surface in , Ann. of Math. (2), 92 (1970), 335-374.
• CHERN, S. S., Deformation of surfaces preserving principal curvature, Differ. Geo. and Complex Anal., H. E. Rauch Memorial volume, Springer-Verlag (1985) 155-1
• ROUSSOS, I. M., The helicoidal surfaces as Bonnet surfaces, Tohoku Math. J. (2) 40 (1988), no. 3, 485– 4
• ROUSSOS, I. M., Tangent developable surfaces as Bonnet surfaces, Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 2, 269–276.
• ROUSSOS, I. M., Global results on Bonnet surfaces, J. Geom. 65 (1999), no. 1-2, 151–168.
• SOYUÇOK, Z., The problem of non-trivial isometries of surfaces preserving principal curvatures, J. Geom. 52 (1995), no. 1-2, 173–188.
• SOYUÇOK, Z., The problem of isometric deformations of a Euclidean hypersurface preserving mean curvature, Bull. Tech. Univ. 49 (1996), no. 3-4, 551–562.
• BAĞDATLI, H., SOYUÇOK, Z., On the problem of isometry of a hypersurface preserving mean curvature, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), no. 1, 49–59.
• CHEN, W., LI, H., On the classification of the timelike Bonnet surfaces, in: Geometry and Topology of Submanifolds, 10, Chern, S. Chen, W., Shelton Street, Covent Garden, London, (1999), 18-31.