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NEW CHARACTERIZATIONS OF SPACELIKE CURVES ON TIMELIKE SURFACES THROUGH THE LINK OF SPECIFIC FRAMES

Year 2018, Volume: 8 Issue: 1.1, 330 - 340, 01.09.2018

Abstract

In this work, considering a regular spacelike curve on a smooth timelike surface in Minkowski 3-space, we investigate relations between the mentioned curve's Darboux and Bishop frames on the timelike surface. Next we obtain Darboux vector of the regular spacelike curve in terms of Bishop apparatus. Thereafter, translating the Darboux vector to the center of the unit sphere, we determine aforementioned spacelike curve. Moreover, we investigate this spherical image's Frenet-Serret and Bishop apparatus and illustrate our results with two examples.

References

  • Akbulut F., (1983) Darboux vectors of the curves on a surface (Turkish). In Ege University, Faculty of Science Conference Series ; 1, 1-40.
  • Biran L., (1970) Differential Geometry Lectures (in Turkish). Istanbul, TR: Istanbul Faculty of Science Publ.
  • Bishop, LR., (1975) There is more than one way to frame a curve. Amer Math Month, 82: 246-251.
  • B¨ukc¨u, B., Karacan, M. K., (2007), The Bishop Darboux rotation axis of the spacelike curve in
  • Minkowski 3-space. Ege University, Journal of the Faculty of Science 3 (1): 1-5. B¨ukc¨u, B., Karacan, M. K., (2007), On the slant helices according to Bishop frame of the timelike curve in Lorentzian space. Tamkang J Math; 39(3): 255-262.
  • B¨ukc¨u, B., Karacan, M. K., (2009), The slant helices according to Bishop frame. Int J Comp Math Sci; 3(2): 67-70.
  • Do Carmo, M., (1976) Differential Geometry of Curves and Surfaces. New Jersey, NJ, USA: Prentice- Hall Inc.
  • Ekici C, Savcı ¨UZ, ¨Unl¨ut¨urk Y., (2013), The relations among instantaneous rotation vectors of a parallel timelike ruled surface. Math Sci Appl E-Notes ; 1 (1): 79-89.
  • Hacısaliho˘glu H. H. (2000), Differential Geometry (in Turkish). Ankara, TR: Faculty of Science Publ.
  • ˙Ilarslan K, Camcı C¸ , Kocayi˘git H, Hacısaliho˘glu H. H. (2003), On the explicit characterization of spherical curves in 3-dimensional Lorentzian space L. J Inverse Ill-Posed Prob; 11 (4): 389-397.
  • Karacan M. K., B¨uk¸c¨u B., (2008), Bishop frame of the timelike curve in Minkowski 3-space. SD ¨U Fen Derg; 3(1): 80-90.
  • Karacan M. K., B¨uk¸c¨u B, Y¨uksel N., (2008), On the dual Bishop Darboux rotation axis of the dual space curve. Appl Sci; 10: 115-120.
  • Lopez R. (2010), Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int Elect Journ Geom., 3(2): 67-101.
  • O’Neill, B., (1983), Semi-Riemannian Geometry with Applications to Relativity. New York, NY, USA: Academic Press.
  • Ratcliffe JG. (2006), Foundations of Hyperbolic Manifolds. New York, NY, USA: Springer Sci- ence+Business Media.
  • U˘gurlu, H. H., C¸ alı¸skan, A. Darboux, (2012), Ani D¨onme Vekt¨orleri ile Spacelike ve Timelike Y¨uzeyler
  • Geometrisi (in Turkish). Manisa, TR: CB ¨U Publ. U˘gurlu H. H., Topal A.,(1996), Relation between Darboux instantaneous rotation vectors of curves on a time-like surfaces. Math Comp Appl., 1(2): 149-157.
  • U˘gurlu H. H., (1999), The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface. Hadronic Journal, 22: 145-155.
  • ¨Unl¨ut¨urk Y, Yılmaz, S., (2016), Smarandache curves of a spacelike curve according to the Bishop frame of type-2, International J.Math. Combin. Vol.4, 29-43.
  • Yılmaz S, Turgut M., (2010), A new version of Bishop frame and an application to spherical images. J Math Anal Appl., 371: 764-776.
Year 2018, Volume: 8 Issue: 1.1, 330 - 340, 01.09.2018

Abstract

References

  • Akbulut F., (1983) Darboux vectors of the curves on a surface (Turkish). In Ege University, Faculty of Science Conference Series ; 1, 1-40.
  • Biran L., (1970) Differential Geometry Lectures (in Turkish). Istanbul, TR: Istanbul Faculty of Science Publ.
  • Bishop, LR., (1975) There is more than one way to frame a curve. Amer Math Month, 82: 246-251.
  • B¨ukc¨u, B., Karacan, M. K., (2007), The Bishop Darboux rotation axis of the spacelike curve in
  • Minkowski 3-space. Ege University, Journal of the Faculty of Science 3 (1): 1-5. B¨ukc¨u, B., Karacan, M. K., (2007), On the slant helices according to Bishop frame of the timelike curve in Lorentzian space. Tamkang J Math; 39(3): 255-262.
  • B¨ukc¨u, B., Karacan, M. K., (2009), The slant helices according to Bishop frame. Int J Comp Math Sci; 3(2): 67-70.
  • Do Carmo, M., (1976) Differential Geometry of Curves and Surfaces. New Jersey, NJ, USA: Prentice- Hall Inc.
  • Ekici C, Savcı ¨UZ, ¨Unl¨ut¨urk Y., (2013), The relations among instantaneous rotation vectors of a parallel timelike ruled surface. Math Sci Appl E-Notes ; 1 (1): 79-89.
  • Hacısaliho˘glu H. H. (2000), Differential Geometry (in Turkish). Ankara, TR: Faculty of Science Publ.
  • ˙Ilarslan K, Camcı C¸ , Kocayi˘git H, Hacısaliho˘glu H. H. (2003), On the explicit characterization of spherical curves in 3-dimensional Lorentzian space L. J Inverse Ill-Posed Prob; 11 (4): 389-397.
  • Karacan M. K., B¨uk¸c¨u B., (2008), Bishop frame of the timelike curve in Minkowski 3-space. SD ¨U Fen Derg; 3(1): 80-90.
  • Karacan M. K., B¨uk¸c¨u B, Y¨uksel N., (2008), On the dual Bishop Darboux rotation axis of the dual space curve. Appl Sci; 10: 115-120.
  • Lopez R. (2010), Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int Elect Journ Geom., 3(2): 67-101.
  • O’Neill, B., (1983), Semi-Riemannian Geometry with Applications to Relativity. New York, NY, USA: Academic Press.
  • Ratcliffe JG. (2006), Foundations of Hyperbolic Manifolds. New York, NY, USA: Springer Sci- ence+Business Media.
  • U˘gurlu, H. H., C¸ alı¸skan, A. Darboux, (2012), Ani D¨onme Vekt¨orleri ile Spacelike ve Timelike Y¨uzeyler
  • Geometrisi (in Turkish). Manisa, TR: CB ¨U Publ. U˘gurlu H. H., Topal A.,(1996), Relation between Darboux instantaneous rotation vectors of curves on a time-like surfaces. Math Comp Appl., 1(2): 149-157.
  • U˘gurlu H. H., (1999), The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface. Hadronic Journal, 22: 145-155.
  • ¨Unl¨ut¨urk Y, Yılmaz, S., (2016), Smarandache curves of a spacelike curve according to the Bishop frame of type-2, International J.Math. Combin. Vol.4, 29-43.
  • Yılmaz S, Turgut M., (2010), A new version of Bishop frame and an application to spherical images. J Math Anal Appl., 371: 764-776.
There are 20 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Y. Ünlütürk This is me

S. Yılmaz This is me

Publication Date September 1, 2018
Published in Issue Year 2018 Volume: 8 Issue: 1.1

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