Research Article
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Minkowski 3-uzayda 2. tip Bishop çatısına göre null olmayan eğrilerin karakterizasyonlarına dair bir inceleme

Year 2016, Volume: 20 Issue: 2, 325 - 335, 01.08.2016
https://doi.org/10.16984/saufenbilder.50911

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

Year 2016, Volume: 20 Issue: 2, 325 - 335, 01.08.2016
https://doi.org/10.16984/saufenbilder.50911

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 . 

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.
There are 16 citations in total.

Details

Subjects Engineering
Journal Section Research Articles
Authors

Süha Yılmaz

Yasin Ünlütürk

Abdullah Mağden

Publication Date August 1, 2016
Submission Date February 17, 2016
Acceptance Date May 12, 2016
Published in Issue Year 2016 Volume: 20 Issue: 2

Cite

APA Yılmaz, S., Ünlütürk, Y., & Mağden, A. (2016). A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space. Sakarya University Journal of Science, 20(2), 325-335. https://doi.org/10.16984/saufenbilder.50911
AMA Yılmaz S, Ünlütürk Y, Mağden A. A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space. SAUJS. August 2016;20(2):325-335. doi:10.16984/saufenbilder.50911
Chicago Yılmaz, Süha, Yasin Ünlütürk, and Abdullah Mağden. “A Study on the Characterizations of Non-Null Curves According to the Bishop Frame of Type-2 in Minkowski 3-Space”. Sakarya University Journal of Science 20, no. 2 (August 2016): 325-35. https://doi.org/10.16984/saufenbilder.50911.
EndNote Yılmaz S, Ünlütürk Y, Mağden A (August 1, 2016) A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space. Sakarya University Journal of Science 20 2 325–335.
IEEE S. Yılmaz, Y. Ünlütürk, and A. Mağden, “A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space”, SAUJS, vol. 20, no. 2, pp. 325–335, 2016, doi: 10.16984/saufenbilder.50911.
ISNAD Yılmaz, Süha et al. “A Study on the Characterizations of Non-Null Curves According to the Bishop Frame of Type-2 in Minkowski 3-Space”. Sakarya University Journal of Science 20/2 (August 2016), 325-335. https://doi.org/10.16984/saufenbilder.50911.
JAMA Yılmaz S, Ünlütürk Y, Mağden A. A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space. SAUJS. 2016;20:325–335.
MLA Yılmaz, Süha et al. “A Study on the Characterizations of Non-Null Curves According to the Bishop Frame of Type-2 in Minkowski 3-Space”. Sakarya University Journal of Science, vol. 20, no. 2, 2016, pp. 325-3, doi:10.16984/saufenbilder.50911.
Vancouver Yılmaz S, Ünlütürk Y, Mağden A. A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space. SAUJS. 2016;20(2):325-3.