Minkowski 3-uzayda 2. tip Bishop çatısına göre null olmayan eğrilerin karakterizasyonlarına dair bir inceleme
Abstract
Bu çalışmada, de Bishop çatısının yeni bir yorumuna göre null olmayan eğrilerin klasik diferensiyel geometrisini inceliyoruz. Bu yeni yorumlanan çatıya, 2. Tip Bishop çatısı şeklinde adlandırıyoruz. Öncelikle, bir adi diferensiyel denklem sistemi elde etmek suretiyle, regüler ve null olmayan eğrilerin konum vektörünü araştırıyoruz. Bu sistemin çözümü, de 2. tip Bishop çatısın göre konum vektörünün bileşenlerini verir. Bununla birlikte, bu yeni çatıya göre birinci, ikinci ve üçüncü mertebeden Bishop düzlemlerini tanımlıyoruz ve bu düzlemlere bağlı olarak de konum vektörlerini karakterize ediyoruz.
References
- A.T. Ali, M.Turgut, “Position vector of a time-like slant helix in Minkowski 3-space”, J Math Anal Appl, vol. 365, no. 1, pp. 559-569, 2010.
- L. R. Bishop, There is more than one way to frame a curve, Amer Math Monthly vol. 82, pp.246-251, 1975.
- B. Y. Chen, “When does the position vector of a space curve always lie in its rectifying plane?”, Amer Math Monthly, vol. 110, pp. 147-152, 2003.
- B. Divjak, “Curves in pseudo-Galilean geometry”, Annales Univ. Sci. Bud, vol. 41, pp. 117-128, 1998.
- K. İlarslan, Ö. Boyacıoğlu, “Position vectors of a time-like and a null helix in Minkowski 3-space”, Chaos Solitons Fractals, vol.38, pp. 1383—1389, 2008.
- R. Lopez, “Differential geometry of curves and surfaces in Lorentz-Minkowski space”, Int. Elec. Journ. Geom. vol. 3, no. 2, pp. 67-101, 2010.
- B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
- E. Özyılmaz, “Classical differential geometry of curves according to type-2 Bishop trihedra”, Math.Comput. Appl., vol. 16, no.4, pp. 858-867, 2011.
- M. Turgut, “On the invariants of time-like dual curves”, Hacettepe Jour. Math. Stat., vol. 37, no. 2, pp. 129-133, 2008.
- M. Turgut, S. Yılmaz, “Contributions to classical differential geometry of the curves in E^3”, Sci. Magna, vol. 4, pp. 5-9, 2008.
- Y. Ünlütürk, S. Yılmaz, “A new version of Bishop frame and its application to Smarandache curves of a spacelike curve in Minkowski 3-space”, to appear.
- S. Yılmaz, M. Turgut, “A new version of Bishop frame and an application to spherical images”, J. Math. Anal. Appl., v. 371, pp. 764-776, 2010.
- S. Yılmaz, “Bishop spherical images of a spacelike curve in Minkowski 3-space”, Int. Jour. Phys. Scie. vol. 5, no. 6, pp. 898-905, 2010.
- S. Yılmaz, “Position vectors of some special space-like curves according to Bishop frame in Minkowski space E_1^3”, Sci. Magna., vol. 5, no. 1, pp. 58-50, 2009.
- S. Yılmaz, E. Özyılmaz, M. Turgut, Ş. Nizamoğlu, “Position vector of a partially null curve derived from a vector differential equation, Int. Sch. Scie. Res. & Innov., vol. 3, no. 11, pp. 846-848, 2009.
- S. Yılmaz , Ü.Z. Savcı, A. Mağden, “Position vector of some special curves in Galilean 3-spaces G_3”, Glo. Jour. Adv. Res. Class. Mod. Geom., vol. 3, no. 1, pp. 7-11, 2014.
A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space
Abstract
In this work, we study classical differential geometry of non-null curves according to the new version of Bishop frame in which we call it along the work as “the Bishop frame of type-2”. First, we investigate position vector of a regular non-null curve by obtaining a system of ordinary differential equations. The solution of the system gives the components of the position vector with respect to the Bishop frame of type-2 in . Moreover, we define the first, second and third order Bishop planes according to this new frame, and also, regardig to these planes, we characterize position vectors in .
Keywords
Spacelike Curve Timelike Curve Position Vector The Bishop Planes The Bishop Frame of Type-2
References
- A.T. Ali, M.Turgut, “Position vector of a time-like slant helix in Minkowski 3-space”, J Math Anal Appl, vol. 365, no. 1, pp. 559-569, 2010.
- L. R. Bishop, There is more than one way to frame a curve, Amer Math Monthly vol. 82, pp.246-251, 1975.
- B. Y. Chen, “When does the position vector of a space curve always lie in its rectifying plane?”, Amer Math Monthly, vol. 110, pp. 147-152, 2003.
- B. Divjak, “Curves in pseudo-Galilean geometry”, Annales Univ. Sci. Bud, vol. 41, pp. 117-128, 1998.
- K. İlarslan, Ö. Boyacıoğlu, “Position vectors of a time-like and a null helix in Minkowski 3-space”, Chaos Solitons Fractals, vol.38, pp. 1383—1389, 2008.
- R. Lopez, “Differential geometry of curves and surfaces in Lorentz-Minkowski space”, Int. Elec. Journ. Geom. vol. 3, no. 2, pp. 67-101, 2010.
- B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
- E. Özyılmaz, “Classical differential geometry of curves according to type-2 Bishop trihedra”, Math.Comput. Appl., vol. 16, no.4, pp. 858-867, 2011.
- M. Turgut, “On the invariants of time-like dual curves”, Hacettepe Jour. Math. Stat., vol. 37, no. 2, pp. 129-133, 2008.
- M. Turgut, S. Yılmaz, “Contributions to classical differential geometry of the curves in E^3”, Sci. Magna, vol. 4, pp. 5-9, 2008.
- Y. Ünlütürk, S. Yılmaz, “A new version of Bishop frame and its application to Smarandache curves of a spacelike curve in Minkowski 3-space”, to appear.
- S. Yılmaz, M. Turgut, “A new version of Bishop frame and an application to spherical images”, J. Math. Anal. Appl., v. 371, pp. 764-776, 2010.
- S. Yılmaz, “Bishop spherical images of a spacelike curve in Minkowski 3-space”, Int. Jour. Phys. Scie. vol. 5, no. 6, pp. 898-905, 2010.
- S. Yılmaz, “Position vectors of some special space-like curves according to Bishop frame in Minkowski space E_1^3”, Sci. Magna., vol. 5, no. 1, pp. 58-50, 2009.
- S. Yılmaz, E. Özyılmaz, M. Turgut, Ş. Nizamoğlu, “Position vector of a partially null curve derived from a vector differential equation, Int. Sch. Scie. Res. & Innov., vol. 3, no. 11, pp. 846-848, 2009.
- S. Yılmaz , Ü.Z. Savcı, A. Mağden, “Position vector of some special curves in Galilean 3-spaces G_3”, Glo. Jour. Adv. Res. Class. Mod. Geom., vol. 3, no. 1, pp. 7-11, 2014.
Details
| Subjects | Engineering |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | August 1, 2016 |
| Submission Date | February 17, 2016 |
| Acceptance Date | May 12, 2016 |
| Published in Issue | Year 2016 Volume: 20 Issue: 2 |
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