Research Article
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Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

Year 2024, Volume: 17 Issue: 2, 447 - 465

Orhan Oğulcan Tuncer

https://doi.org/10.36890/iejg.1440270

Abstract

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Keywords

Bertrand curves Mannheim curves framed curves singularity non-flat space forms Hopf map

References

  • [1] Altın Erdem, H., İlarslan, K.: Spacelike Bertrand curves in Minkowski 3-space revisited. An. ¸St. Univ. Ovidius Constanta 31 (3), 87-109 (2023).
  • [2] Balgetir, H., Bekta¸s, M., Inoguchi, J.: Null Bertrand curves in Minkowski 3-space and their characterizations. Note Mat. 23 (1), 7-13 (2004/05).
  • [3] Benyounes, M., Loubeau, E., Nishikawa, S.: Generalized Cheeger-Gromoll metrics and the Hopf map. Diff. Geom. Appl. 39, 187-213 (2011).
  • [4] Bertrand, J.: Mémoire sur la théorie des courbes é double courbure. Comptes Rendus 36 (1850); Journal de Mathématiques Pures et Appliquées 15, 332-350 (1850).
  • [5] Camcı, Ç., Uçum, A., ˙Ilarslan, K.: A new approach to Bertrand curves in Euclidean 3-space. J. Geom 111, 49, (2020).
  • [6] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane?. Amer. Math. Monthly 110, 147-152 (2003).
  • [7] Chen, B. Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Acad. Sin. 33, 77–90 (2005).
  • [8] Chen, L., Takahashi, M.: Dualities and evolutes of fronts in hyperbolic and de Sitter space. J. Math. Anal. Appl. 437, 133-159 (2016).
  • [9] Cheng, Y.-M., Lin, C.-C.: On the generalized Bertrand curves in Euclidean N-spaces. Note Mat. 29 (2), 33-39 (2009).
  • [10] Choi, J. H., Kang, T. H., Kim, Y. H.: Bertrand curves in 3-dimensional space forms. Appl. Math. Comput. 219, 1040–1046 (2012).
  • [11] Choi, J. H., Kang, T. H., Kim, Y. H.: Mannheim curves in 3-dimensional space forms. Bull. Korean Math. Soc. 50, 1099–1108 (2013).
  • [12] Ekmekci, N., Ilarslan, K.: On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst. 3 (2), 17-24 (2001).
  • [13] Ersoy, S., Tosun, M., Matsuda, H.: Generalized Mannheim curves in Minkowski space-time E4 1 . Hokkaido Math. J. 41 (3), 441-461 (2012).
  • [14] Fukunaga, T., Takahashi, M.: Existence and Uniqueness for Legendre Curves. J. Geom. 104, 297-307 (2013).
  • [15] Grbovic, M., İlarslan, K., Nesovic, E.: On null and pseudo null Mannheim curves in Minkowski 3-space. J. Geom. 105, 177–183 (2014).
  • [16] Gök, I, Okuyucu, O.Z., Ekmekci, N., Yaylı, Y.: On Mannheim partner curves in three dimensional Lie groups. Miskolc Math. 15 (2), 467-479 (2014).
  • [17] Honda, S., Takahashi, M.: Framed curves in the Euclidean space. Adv. Geom. 16 (3), 265–276 (2016).
  • [18] Honda, S., Takahashi, M.: Evolutes and focal surfaces of framed immersions in the Euclidean space. Proc. Roy. Soc. Edinburgh Sect. A 150, 497-516 (2020).
  • [19] Honda, S., Takahashi, M.: Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space. Turk. J. Math. 44, 883–899 (2020).
  • [20] Honda, S., Takahashi, M., Haiou, Y.: Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space. J. Geom 114:12, (2023).
  • [21] Huang, J., Chen, L., Izumiya, S., Pei, D.: Geometry of special curves and surfaces in 3-space form. J. Geom. Phys. 136, 31-38 (2019).
  • [22] Huang, J., Pei, D.: Singular Special Curves in 3-Space Forms. Mathematics 8, 846, (2020).
  • [23] İlarslan, K., Kılıç Aslan, N.: On generalized null Mannheim curves in E4 2 . Math. Meth. Appl. Sci. 44 (9), 7588-7600 (2021).
  • [24] Li, Y., Pei, D.: Pedal Curves of Fronts in the sphere. J. Nonlinear Sci. Appl. 9, 836-844 (2016).
  • [25] Li, P., Pei, D., Zhao, X.: Spacelike Framed Curves with Lightlike Components and Singularities of Their Evolutes and Focal Surfaces in Minkowski 3-space. Acta Math. Sin. Engl. Ser., DOI: 10.1007/s10114-023-1672-2, (2023).
  • [26] Li, Y., Sun, Q-Y.: Evolutes of fronts in the Minkowski plane. Math. Meth. Appl. Sci. 42 (16), 5416-5426 (2019).
  • [27] Li, Y., Tuncer, O.O.: On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space. Math. Meth. Appl. Sci. 46 (9), 11157-11171 (2023).
  • [28] Li, Y., Uçum, A., İlarslan, K., Camcı, Ç.: A New Class of Bertrand Curves in Euclidean 4-Space. Symmetry 14 (6), 1191, (2022).
  • [29] Liu, H., Wang, F.: Mannheim partner curves in 3-space. J. Geom. 88, 120-126 (2008).
  • [30] Lucas, P., Ortega-Yagües, J.A.: Bertrand curves in the three-dimensional sphere. J. Geom. Phys. 62, 1903-1914 (2012).
  • [31] Lucas, P., Ortega-Yagües, J. A.: Bertrand curves in non-flat 3-dimensional (Riemannian or Lorentzian) space forms. Bull. Korean Math. Soc. 50, 1109-1126 (2013).
  • [32] Lucas, P., Ortega-Yagües, J. A.: Rectifying curves in the three-dimensional sphere. J. Math. Anal. Appl. 421 (2), 1855-1868 (2015).
  • [33] Lucas, P., Ortega-Yagües, J. A.: Rectifying Curves in the Three-Dimensional Hyperbolic Space. Mediterr. J. Math. 13, 2199-2214 (2016).
  • [34] Lyons, D. W.: An elementary introduction to the Hopf fibration. Math. Mag. 76 (2), 87-98 (2003).
  • [35] Matsuda, H., Yorozu, S.: Notes on Bertrand curves. Yokohama Math. J. 50 (1-2), 41-58 (2003).
  • [36] Okuyucu, O. Z., Gök, İ, Yaylı, Y., Ekmekci, N.: Bertrand curves in three dimensional Lie groups. Miskolc Math. 17 (2), 999–1010 (2017).
  • [37] Pears, L. R.: Bertrand Curves in Riemannian Space. J. London Math. Soc. 10, 180-183 (1935).
  • [38] Saint-Venant, J. C.: Mémoire sur les lignes courbes non planes. Journal d’Ecole Polytechnique 30, 1-76 (1845).
  • [39] Takahashi, M., Yu, H.: Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere, Mathematics 10, 1292, (2022).
  • [40] Tuncer, O. O.: Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space. arXiv:2304.08045, (2023).
  • [41] Tuncer, O. O., Ceyhan, H., Gök, ˙I, Ekmekci, F.N.: Notes on pedal and contrapedal curves of fronts in the Euclidean plane. Math. Meth. Appl. Sci. 41, 5096-5111 (2018).
  • [42] Tuncer, O. O., Gök, ˙I: Hyperbolic caustics of light rays reflected by hyperbolic front mirrors. Eur. Phys. J. Plus 138:266 (2023).
  • [43] Uçum, A., Camcı, Ç., ˙Ilarslan, K.: A New Approach to Mannheim Curve in Euclidean 3-Space. Tamkang J. Math. 54, (2021).
  • [44] Uçum, A., İlarslan, K.: On timelike Bertrand curves in Minkowski 3-space. Honam Math. J. 38 (3), 467-477 (2016).
  • [45] Wang, Y., Chang, Y.: Mannheim curves and spherical curves. Int. J. Geom. Methods Mod. Phys. 17 (7), 2050101, (2020).
  • [46] Wang, Y., Pei, D., Gao, R.: Generic Properties of Framed Rectifying Curves. Mathematics 7 (1), 37, (2019).
  • [47] Yu, H., Pei, D., Cui, X.: Evolutes of Fronts on Euclidean 2-sphere. J. Nonlinear Sci. Appl. 8, 678-686 (2015).
  • [48] Zhao, W., Pei, D., Cao, X.: Mannheim Curves in Nonflat 3-Dimensional Space Forms. Adv. Math. Phys. 2015, Article ID 319046, (2015).

Year 2024, Volume: 17 Issue: 2, 447 - 465

Orhan Oğulcan Tuncer

https://doi.org/10.36890/iejg.1440270

Abstract

References

  • [1] Altın Erdem, H., İlarslan, K.: Spacelike Bertrand curves in Minkowski 3-space revisited. An. ¸St. Univ. Ovidius Constanta 31 (3), 87-109 (2023).
  • [2] Balgetir, H., Bekta¸s, M., Inoguchi, J.: Null Bertrand curves in Minkowski 3-space and their characterizations. Note Mat. 23 (1), 7-13 (2004/05).
  • [3] Benyounes, M., Loubeau, E., Nishikawa, S.: Generalized Cheeger-Gromoll metrics and the Hopf map. Diff. Geom. Appl. 39, 187-213 (2011).
  • [4] Bertrand, J.: Mémoire sur la théorie des courbes é double courbure. Comptes Rendus 36 (1850); Journal de Mathématiques Pures et Appliquées 15, 332-350 (1850).
  • [5] Camcı, Ç., Uçum, A., ˙Ilarslan, K.: A new approach to Bertrand curves in Euclidean 3-space. J. Geom 111, 49, (2020).
  • [6] Chen, B. Y.: When does the position vector of a space curve always lie in its rectifying plane?. Amer. Math. Monthly 110, 147-152 (2003).
  • [7] Chen, B. Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Acad. Sin. 33, 77–90 (2005).
  • [8] Chen, L., Takahashi, M.: Dualities and evolutes of fronts in hyperbolic and de Sitter space. J. Math. Anal. Appl. 437, 133-159 (2016).
  • [9] Cheng, Y.-M., Lin, C.-C.: On the generalized Bertrand curves in Euclidean N-spaces. Note Mat. 29 (2), 33-39 (2009).
  • [10] Choi, J. H., Kang, T. H., Kim, Y. H.: Bertrand curves in 3-dimensional space forms. Appl. Math. Comput. 219, 1040–1046 (2012).
  • [11] Choi, J. H., Kang, T. H., Kim, Y. H.: Mannheim curves in 3-dimensional space forms. Bull. Korean Math. Soc. 50, 1099–1108 (2013).
  • [12] Ekmekci, N., Ilarslan, K.: On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst. 3 (2), 17-24 (2001).
  • [13] Ersoy, S., Tosun, M., Matsuda, H.: Generalized Mannheim curves in Minkowski space-time E4 1 . Hokkaido Math. J. 41 (3), 441-461 (2012).
  • [14] Fukunaga, T., Takahashi, M.: Existence and Uniqueness for Legendre Curves. J. Geom. 104, 297-307 (2013).
  • [15] Grbovic, M., İlarslan, K., Nesovic, E.: On null and pseudo null Mannheim curves in Minkowski 3-space. J. Geom. 105, 177–183 (2014).
  • [16] Gök, I, Okuyucu, O.Z., Ekmekci, N., Yaylı, Y.: On Mannheim partner curves in three dimensional Lie groups. Miskolc Math. 15 (2), 467-479 (2014).
  • [17] Honda, S., Takahashi, M.: Framed curves in the Euclidean space. Adv. Geom. 16 (3), 265–276 (2016).
  • [18] Honda, S., Takahashi, M.: Evolutes and focal surfaces of framed immersions in the Euclidean space. Proc. Roy. Soc. Edinburgh Sect. A 150, 497-516 (2020).
  • [19] Honda, S., Takahashi, M.: Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space. Turk. J. Math. 44, 883–899 (2020).
  • [20] Honda, S., Takahashi, M., Haiou, Y.: Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space. J. Geom 114:12, (2023).
  • [21] Huang, J., Chen, L., Izumiya, S., Pei, D.: Geometry of special curves and surfaces in 3-space form. J. Geom. Phys. 136, 31-38 (2019).
  • [22] Huang, J., Pei, D.: Singular Special Curves in 3-Space Forms. Mathematics 8, 846, (2020).
  • [23] İlarslan, K., Kılıç Aslan, N.: On generalized null Mannheim curves in E4 2 . Math. Meth. Appl. Sci. 44 (9), 7588-7600 (2021).
  • [24] Li, Y., Pei, D.: Pedal Curves of Fronts in the sphere. J. Nonlinear Sci. Appl. 9, 836-844 (2016).
  • [25] Li, P., Pei, D., Zhao, X.: Spacelike Framed Curves with Lightlike Components and Singularities of Their Evolutes and Focal Surfaces in Minkowski 3-space. Acta Math. Sin. Engl. Ser., DOI: 10.1007/s10114-023-1672-2, (2023).
  • [26] Li, Y., Sun, Q-Y.: Evolutes of fronts in the Minkowski plane. Math. Meth. Appl. Sci. 42 (16), 5416-5426 (2019).
  • [27] Li, Y., Tuncer, O.O.: On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space. Math. Meth. Appl. Sci. 46 (9), 11157-11171 (2023).
  • [28] Li, Y., Uçum, A., İlarslan, K., Camcı, Ç.: A New Class of Bertrand Curves in Euclidean 4-Space. Symmetry 14 (6), 1191, (2022).
  • [29] Liu, H., Wang, F.: Mannheim partner curves in 3-space. J. Geom. 88, 120-126 (2008).
  • [30] Lucas, P., Ortega-Yagües, J.A.: Bertrand curves in the three-dimensional sphere. J. Geom. Phys. 62, 1903-1914 (2012).
  • [31] Lucas, P., Ortega-Yagües, J. A.: Bertrand curves in non-flat 3-dimensional (Riemannian or Lorentzian) space forms. Bull. Korean Math. Soc. 50, 1109-1126 (2013).
  • [32] Lucas, P., Ortega-Yagües, J. A.: Rectifying curves in the three-dimensional sphere. J. Math. Anal. Appl. 421 (2), 1855-1868 (2015).
  • [33] Lucas, P., Ortega-Yagües, J. A.: Rectifying Curves in the Three-Dimensional Hyperbolic Space. Mediterr. J. Math. 13, 2199-2214 (2016).
  • [34] Lyons, D. W.: An elementary introduction to the Hopf fibration. Math. Mag. 76 (2), 87-98 (2003).
  • [35] Matsuda, H., Yorozu, S.: Notes on Bertrand curves. Yokohama Math. J. 50 (1-2), 41-58 (2003).
  • [36] Okuyucu, O. Z., Gök, İ, Yaylı, Y., Ekmekci, N.: Bertrand curves in three dimensional Lie groups. Miskolc Math. 17 (2), 999–1010 (2017).
  • [37] Pears, L. R.: Bertrand Curves in Riemannian Space. J. London Math. Soc. 10, 180-183 (1935).
  • [38] Saint-Venant, J. C.: Mémoire sur les lignes courbes non planes. Journal d’Ecole Polytechnique 30, 1-76 (1845).
  • [39] Takahashi, M., Yu, H.: Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere, Mathematics 10, 1292, (2022).
  • [40] Tuncer, O. O.: Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space. arXiv:2304.08045, (2023).
  • [41] Tuncer, O. O., Ceyhan, H., Gök, ˙I, Ekmekci, F.N.: Notes on pedal and contrapedal curves of fronts in the Euclidean plane. Math. Meth. Appl. Sci. 41, 5096-5111 (2018).
  • [42] Tuncer, O. O., Gök, ˙I: Hyperbolic caustics of light rays reflected by hyperbolic front mirrors. Eur. Phys. J. Plus 138:266 (2023).
  • [43] Uçum, A., Camcı, Ç., ˙Ilarslan, K.: A New Approach to Mannheim Curve in Euclidean 3-Space. Tamkang J. Math. 54, (2021).
  • [44] Uçum, A., İlarslan, K.: On timelike Bertrand curves in Minkowski 3-space. Honam Math. J. 38 (3), 467-477 (2016).
  • [45] Wang, Y., Chang, Y.: Mannheim curves and spherical curves. Int. J. Geom. Methods Mod. Phys. 17 (7), 2050101, (2020).
  • [46] Wang, Y., Pei, D., Gao, R.: Generic Properties of Framed Rectifying Curves. Mathematics 7 (1), 37, (2019).
  • [47] Yu, H., Pei, D., Cui, X.: Evolutes of Fronts on Euclidean 2-sphere. J. Nonlinear Sci. Appl. 8, 678-686 (2015).
  • [48] Zhao, W., Pei, D., Cao, X.: Mannheim Curves in Nonflat 3-Dimensional Space Forms. Adv. Math. Phys. 2015, Article ID 319046, (2015).
There are 48 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Orhan Oğulcan Tuncer 0000-0002-2916-1380

Early Pub Date September 20, 2024
Publication Date
Submission Date February 20, 2024
Acceptance Date May 24, 2024
Published in Issue Year 2024 Volume: 17 Issue: 2

Cite

APA Tuncer, O. O. (2024). Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures. International Electronic Journal of Geometry, 17(2), 447-465. https://doi.org/10.36890/iejg.1440270
AMA Tuncer OO. Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures. Int. Electron. J. Geom. September 2024;17(2):447-465. doi:10.36890/iejg.1440270
Chicago Tuncer, Orhan Oğulcan. “Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-Zero Constant Curvatures”. International Electronic Journal of Geometry 17, no. 2 (September 2024): 447-65. https://doi.org/10.36890/iejg.1440270.
EndNote Tuncer OO (September 1, 2024) Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures. International Electronic Journal of Geometry 17 2 447–465.
IEEE O. O. Tuncer, “Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures”, Int. Electron. J. Geom., vol. 17, no. 2, pp. 447–465, 2024, doi: 10.36890/iejg.1440270.
ISNAD Tuncer, Orhan Oğulcan. “Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-Zero Constant Curvatures”. International Electronic Journal of Geometry 17/2 (September 2024), 447-465. https://doi.org/10.36890/iejg.1440270.
JAMA Tuncer OO. Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures. Int. Electron. J. Geom. 2024;17:447–465.
MLA Tuncer, Orhan Oğulcan. “Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-Zero Constant Curvatures”. International Electronic Journal of Geometry, vol. 17, no. 2, 2024, pp. 447-65, doi:10.36890/iejg.1440270.
Vancouver Tuncer OO. Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures. Int. Electron. J. Geom. 2024;17(2):447-65.
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