Bertrand curves in non-flat 3-dimensional (Riemannian or Lorentzian) space forms
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let M3 q(c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c 6= 0. A curve α immersed in M3 q(c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: nonnull Bertrand curves in M3 q(c) correspond with curves for which there exist two constants λ 6= 0 and μ such that λκ + μ = 1, where τ and stand for the curvature and torsion of the curve. As a consequence, non-null helices in M3 q(c) are the only twisted curves in M3 q(c) having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.
Abstract
Let M3 q(c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c 6= 0. A curve α immersed in M3 q(c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: nonnull Bertrand curves in M3 q(c) correspond with curves for which there exist two constants λ 6= 0 and μ such that λκ + μ = 1, where τ and stand for the curvature and torsion of the curve. As a consequence, non-null helices in M3 q(c) are the only twisted curves in M3 q(c) having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.
참고문헌 (23)
- Y. A. Aminov, "Differential Geometry and Topology of Curves" (2000), Gordon and Breach Science Publishers
- M. Barros, "General helices and a theorem of Lancret", Proc. Amer. Math. Soc, 125(5), pp.125~1997 (1997)
- M. Barros, "Solutions of the Betchov-Da Rios soliton equation: A Lorentzian approach", J. Geom. Phys, 31(2), pp.217~228 (1999)
- M. Barros, "General helices in the 3-dimensional Lorentzian space forms", Rocky Mountain J. Math, 31(2), pp.373~388 (2001)
- J. Bertrand, "Memoire sur la theorie des courbes e double courbure", Comptes Rendus, 36, pp.332~350 (1850)
- G. S. Birman, "Trigonometry in Lorentzian geometry", Amer. Math. Monthly, 91(9), pp.543~549 (1984)
- G. S. Birman, "Gauss-Bonnet theorem for 2-dimensional spacetimes", Michigan Math. J, 31(1), pp.77~81 (1984)
- Y.-M. Cheng, "On the generalized Bertrand curves in Euclidean N-spaces", Note Mat, 29(2), pp.33~39 (2009)
- K. L. Duggal, "Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications" (1996), Kluwer Academic Publishers
- N. Ekmekci, "On Bertrand curves and their characterization", Differ. Geom. Dyn. Syst, 3(2), pp.17~24 (2001)
- S. Ersoy, "Timelike Bertrand Curves in Semi-Euclidean Space", arXiv:1003.1220v1, ~
- A. Ferrandez, "Null helices in Lorentzian space forms", Internat. J. Modern Phys. A, 16(30), pp.4845~4863 (2001)
- A. Gorgulu, "A generalization of the Bertrand curves as general inclined curves in En", Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist, 35(1), pp.53~60 (1986)
- A. Gray, "Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapter 3.5, Clothoids", pp.64~66 (1997), CRC Press
- G. Harary, "3D Euler spirals for 3D curve completion", Proceedings of the 2010 Annual Symposium on Computational Geometry SoCG’10, pp.393~402 (2010)
- M. Kulahc, "Bertrand curves of AW(k)-type in Lorentzian space", Nonlinear Anal, 70(4), pp.1725~1731 (2009)
- H. F. Lai, "Weakened Bertrand curves", Tohoku Math. J, 19, pp.141~155 (1967)
- H. Matsuda, "Notes on Bertrand curves", Yokohama Math. J, 50(1), pp.41~58 (2003)
- E. Nessovic, "Curves in Lorentzian spaces", Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat, 8(3), pp.685~696 (2005)
- H. B. Oztekin, "Weakened Bertrand curves in the Galilean space G3", J. Adv. Math. Stud, 2(2), pp.69~76 (2009)
- L. R. Pears, "Bertrand curves in Riemannian space", J. London Math. Soc, 1(3), pp.180~183 (1935)
- J. C. Saint-Venant, "Memoire sur les lignes courbes non planes", Journal d’Ecole Polytechnique, 30, pp.1~76 (1845)
- M. Y. Yilmaz, "General properties of Bertrand curves in Riemann-Otsuki space", Nonlinear Anal, 69(10), pp.3225~3231 (2008)