Research Article
Year 2019,
, 87 - 101, 22.03.2019
Abstract
In this paper, we investigate the surfaces generated by
binormal motion of Bertrand curves, which is called Razzaboni surface, in
Minkowski 3-space. We discussed the geometric properties of these surfaces in with respect to
the character of Bertrand geodesics. Then, we define the Razzaboni
transformation for a given Razzaboni surface. In other words, we prove that
there exists a dual of Razzaboni surface for each case. Finally, we show that
Razzaboni transformation maps the surface which has
Bertrand geodesic with constant curvature, to the surface whose Bertrand
geodesic also has constant curvature with opposite sign.
binormal motion of Bertrand curves, which is called Razzaboni surface, in
Minkowski 3-space. We discussed the geometric properties of these surfaces in with respect to
the character of Bertrand geodesics. Then, we define the Razzaboni
transformation for a given Razzaboni surface. In other words, we prove that
there exists a dual of Razzaboni surface for each case. Finally, we show that
Razzaboni transformation maps the surface which has
Bertrand geodesic with constant curvature, to the surface whose Bertrand
geodesic also has constant curvature with opposite sign.
Keywords
References
- [1]. Eisenhart L.P., A Treatise on the Differential Geometry of Curves and Surfaces. Dover, New York (1960).
- [2]. Lopez R. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, arXiv:0810.3351v1, (2008).
- [3]. Razzaboni A., Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand. Bologna Mem., 10 (1903) 539-548.
- [4]. Schief W.K., On the Integrability of Bertrand curves and Razzaboni surfaces, Journal of Geometry and Physics, 45 (2002) 130-150.
- [5]. Fibbi C., Sulle superficie che contengono un sistema di geodetiche a torsione constante, Pisa Ann., 5 (1888) 79-164.
- [6]. Razzaboni A., Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand. Bologna, Tipografia Gamberini e Parmeggiani (1998).
- [7]. Rogers C. and Schief W.K., Backlund and Darboux Transformations, Geometry of Modern Applications in Soliton Theory, Cambridge University Press, (2002).
- [8]. Schief W.K. and Rogers C., Binormal Motion of Curves of Constant Curvature and Torsion, Generation of Soliton Surfaces, Proc. R. Soc. Lond. A., 455 (1999) 3163-3188.
- [9]. Gürbüz N., The Motion of Timelike Surfaces in Timelike Geodesic Coordinates, Int. Journal of Math. Analysis, 4 (2010) 349-356.
- [10]. Ding Q.and Inoguchi J., Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons and Fractals, 21 (2004) 669-677.
- [11]. Inoguchi J., Biharmonic curves in Minkowski 3-space, International Journal of Mathematics and Mathematical Sciences, 21 (2003) 1365-1368.
- [12]. Erdoğdu M. and Özdemir M., Geometry of Hasimoto Surfaces in Minkowski 3-space. Mathematical Physics, Analysis and Geometry, 17 (2014) 169-281.
- [13]. Inoguchi J. Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo Journal of Mathematics, 21 (1998) 141-152.
- [14]. Özdemir M. and Ergin A.A., Parallel Frames of Non-Lightlike Curves. Missouri Journal of Mathematical Sciences, 20 (2008) 127-137.
Year 2019,
, 87 - 101, 22.03.2019
Abstract
Bu çalışmada, Minkowski-3 uzayında, Bertrand eğrilerin binormal hareketi
ile meydana gelen, Razzaboni yüzeyi adı verilen yüzeyler incelenmiştir.
Minkowski-3 uzayındaki bu yüzeylerin geometrik özelliklerini Bertrand geodeziklerin
karakterine bağlı olarak tartıştık. Daha sonra, verilen bir Razzaboni yüzeyi
için Razzaboni dönüşümünü tanımladık. Diğer bir deyişle, her durum için
Razzaboni yüzeyinin bir duali olduğunu ispatladık. Son olarak, Razzaboni
dönüşümlerinin; sabit eğrilikli Bertrand geodeziğe sahip yüzeyini; işareti ters olmak
üzere aynı sabit eğrilikli Bertrand geodeziğe sahip yüzeyine dönüştürdüğünü
gösterdik.
Keywords
References
- [1]. Eisenhart L.P., A Treatise on the Differential Geometry of Curves and Surfaces. Dover, New York (1960).
- [2]. Lopez R. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space, arXiv:0810.3351v1, (2008).
- [3]. Razzaboni A., Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand. Bologna Mem., 10 (1903) 539-548.
- [4]. Schief W.K., On the Integrability of Bertrand curves and Razzaboni surfaces, Journal of Geometry and Physics, 45 (2002) 130-150.
- [5]. Fibbi C., Sulle superficie che contengono un sistema di geodetiche a torsione constante, Pisa Ann., 5 (1888) 79-164.
- [6]. Razzaboni A., Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand. Bologna, Tipografia Gamberini e Parmeggiani (1998).
- [7]. Rogers C. and Schief W.K., Backlund and Darboux Transformations, Geometry of Modern Applications in Soliton Theory, Cambridge University Press, (2002).
- [8]. Schief W.K. and Rogers C., Binormal Motion of Curves of Constant Curvature and Torsion, Generation of Soliton Surfaces, Proc. R. Soc. Lond. A., 455 (1999) 3163-3188.
- [9]. Gürbüz N., The Motion of Timelike Surfaces in Timelike Geodesic Coordinates, Int. Journal of Math. Analysis, 4 (2010) 349-356.
- [10]. Ding Q.and Inoguchi J., Schrödinger flows, binormal motion for curves and second AKNS-hierarchies, Chaos Solitons and Fractals, 21 (2004) 669-677.
- [11]. Inoguchi J., Biharmonic curves in Minkowski 3-space, International Journal of Mathematics and Mathematical Sciences, 21 (2003) 1365-1368.
- [12]. Erdoğdu M. and Özdemir M., Geometry of Hasimoto Surfaces in Minkowski 3-space. Mathematical Physics, Analysis and Geometry, 17 (2014) 169-281.
- [13]. Inoguchi J. Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo Journal of Mathematics, 21 (1998) 141-152.
- [14]. Özdemir M. and Ergin A.A., Parallel Frames of Non-Lightlike Curves. Missouri Journal of Mathematical Sciences, 20 (2008) 127-137.
There are 14 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 22, 2019 |
| Submission Date | September 19, 2018 |
| Acceptance Date | January 4, 2019 |
| Published in Issue | Year 2019 |