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A New Insight on Rectifying-Type Curves in Euclidean 4-Space

Year 2023, Volume: 16 Issue: 2, 644 - 652, 29.10.2023

Zehra İşbilir , Murat Tosun

https://doi.org/10.36890/iejg.1291893
Cited By: 3

Abstract

In this study, our purpose is to determine the generalized rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space $E_4$. Also, some characterizations of them are given. We construct some correlations between curvatures and invariants of generalized rectifying-type curves. Additionally, we obtain an illustrative example with respect to the rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space $E_4$.

Keywords

Myller configuration generalized rectifying type curves four-dimensional space

Supporting Institution

-

Project Number

-

Thanks

The authors would like to thank the editors and anonymous referees for their invaluable comments. Zehra İşbilir and Murat Tosun

References

  • [1] Akyiğit, M., Yıldız, Ö. G.: On the framed normal curves in Euclidean 4-space. Fundam. J. Math. Appl. 4 (4), 258–263 (2021).
  • [2] Breuer, S., Gottlieb, D.: Explicit characterization of spherical curves. Proc. Amer. Math. Soc. 27 (1), 126–127 (1971).
  • [3] Chen, B.-Y.: When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Monthly. 110 (2), 147–152 (2003).
  • [4] Chen, B.-Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48 (2), 209-–214 (2017).
  • [5] Chen, B.-Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Academia Sinica. 33 (2), 77–90 (2005).
  • [6] Constantinescu, O.: Myller configurations in Finsler spaces. Differential Geometry-Dynamical Systems. 8, 69–76 (2006).
  • [7] Deshmukh, S., Chen, B.-Y., Alshammari, S. H.: On rectifying curves in Euclidean 3-space. Turk. J. Math. 42 (2), 609-–620 (2018).
  • [8] Doğan Yazıcı, B., Özkaldı Karaku¸s, S., Tosun, M.: Characterizations of framed curves in four-dimensional Euclidean space. Univers. J. Math. Appl. 4 (4), 125–131 (2021).
  • [9] Gökçelik, F., Bozkurt, Z., Gök, İ, Ekmekçi, F. N., Yaylı, Y.: Parallel transport frame in 4-dimensional Euclidean space. Caspian J. Math. Sci. 3 (1), 91–103 (2014).
  • [10] Gluck, H.: Higher curvatures of curves in Euclidean space. Amer. Math. Monthly. 73, 699–704 (1966).
  • [11] Heroiu, B.: Versor fields along a curve in a four dimensional Lorentz space. J. Adv. Math. Stud. 4 (1), 49–57 (2011).
  • [12] İlarslan, K.: Spacelike normal curves in Minkowski space E31 . Turk. J. Math. 29 (2), 53–63 (2005).
  • [13] İlarslan, K., Nešovic, E.: Spacelike and timelike normal curves in Minkowski space-time. Publications de l’Institut Mathmatique. 85 (99), 111–118 (2009).
  • [14] İlarslan, K., Nešovic, E.: Some characterizations of rectifying curves in Euclidean 4-spaces E4. Turk. J. Math. 32, 21–30 (2008).
  • [15] İlarslan, K., Nešovic, E.: Some characterizations of osculating curves in the Euclidean spaces. Demonstratio Mathematica. 41 (4), 931–940 (2008).
  • [16] İlarslan, K., Nešovic, E.: The first kind and the second kind osculating curves in Minkowski space-time. Compt. Rend. Acad. Bulg. Sci. 62 (6), 677–686 (2009).
  • [17] İlarslan, K., Nešovic, E., Petrovic–Torgašev, M.: Some characterizations of rectifying curves in the Minkowski 3–space. Novi Sad J. Math.. 33 (2), 23-–32 (2003).
  • [18] İlarslan, K., Nešovic, E.: Timelike and null normal curves in Minkowski space E31 . Indian J. Pure Appl. Math. 35 (7), 881–888 (2004).
  • [19]İlarslan, K., Nešovic, E.: Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time. Taiwan. J. Math. 12 (5), 1035–1044 (2008).
  • 20] İlarslan, K., Nešovic, E.: Spacelike and timelike normal curves in Minkowski space-time. Publications de l’Institut Mathématique. 85 (99), 111–118 (2009).
  • [21] İlarslan, K., Nešovi´c, E.: Some characterizations of pseudo and partially null osculating curves in Minkowski space-time. Int. Electron. J. Geom. 4 (2), 1–12 (2011).
  • [22] İlarslan, K., Nešovic, E.: On rectifying curves as centrodes and extremal curves in the Minkowski 3-space. Novi Sad J. Math.. 37 (1), 53–64 (2007).
  • [23] İşbilir, Z., Tosun, M.: On generalized osculating-type curves in Myller configuration. An. Şt. Univ. Ovidius Constanta, Ser. Mat. XXXII (2), (2024).
  • [24] İşbilir, Z., Tosun, M.: An extended framework for osculating-type curves in four-dimensional Euclidean space. (2023) (submitted).
  • [25] Keskin, Ö., Yaylı, Y.: Rectifying-type curves and rotation minimizing frame Rn. arXiv preprint, (2019). https://doi.org/10.48550/ arXiv.1905.04540.
  • [26] Kişi, ˙I.: Some characterizations of canal surfaces in the four dimensional Euclidean space, Ph.D. Thesis, Kocaeli University, (2018).
  • [27] Kuhnel, W.: Differential Geometry: Curves-Surfaces-Manifolds, Braunschweig, Wiesbaden (1999).
  • [28] Levi-Civita, T.: Rendiconti del Circolo di Palermo. (XLII), 173 (1917).
  • [29] Levi-Civita, T.: Lezioni di calcolo differentiate assoluto, Zanichelli, (1925).
  • [30] Macsim, G., Mihai, A., Olteanu, A.: Curves in a Myller configuration. International Conference on Applied and Pure Mathematics (ICAPM 2017), Iaşi, November 2-5, 2017.
  • [31] Macsim, G., Mihai, A., Olteanu, A.: On rectifying-type curves in a Myller configuration. Bull. Korean Math. Soc. 56 (2), 383–390 (2019).
  • [32] Macsim, G., Mihai, A., Olteanu, A.: Special curves in a Myller configuration. Proceedings of the 16th Workshop on Mathematics, Computer Science and Technical Education, Department of Mathematics and Computer Science, Volume 2, (2019).
  • [33] Miron, R.: The Geometry of Myller Configurations. Applications to Theory of Surfaces and Nonholonomic Manifolds, Romanian Academy, (2010).
  • [34] Miron, R.: Geometria unor configuratii Myller. Analele ¸St. Univ. VI (3), (1960).
  • [35] Miron, R.: Myller configurations and Vranceanu nonholonomic manifolds. Scientific Studies and Research. 21 (1), (2011).
  • [36] Miron, R.: Configura¸tii Myller M(C, ξi1, Tn−1) în spa¸tii Riemann Vn, Aplica¸tii la studiul hipersuprafe¸telor din Vn. Studii şi Cerc. Şt. Mat. Iaşi. XII (1), (1962).
  • [37] Miron, R.: Configuratii Myller M(C, ξ1 i , Tm) în spa¸tii Riemann Vn. Aplica¸tii la studiul varietˇa¸tilor Vm din Vn. Studii ¸si Cerc. St. Mat. Ia¸si. XIII (1), (1962).
  • [38] Miron, R.: Les configurations de MyllerM(C, ξ1i , Tn−1) dans les espaces de Riemann Vn (I). Tensor. 12 (3), (1962) (Japonia).
  • [39] Miron, R.: Les configurations de MyllerM(C, ξi1, Tm) dans les espaces de Riemann Vn. Tensor. V (1), (1964) (Japonia).
  • [40] Miron, R., Pop, I.: Topologie algebricˇa: omologie, omotopie, spa¸tii de acoperire. Ed. Academiei, (1974).
  • [41] Miron, R., Branzei, D.: Backgrounds of arithmetic and geometry, World Scientific Publishing, S. Pure Math. 23, (1995).
  • [42] Miron, R.: Geometria Configura¸tiilor Myller. Editura Tehnicˇa. Bucure¸sti (1966).
  • [43] Otsuki, T.: Differential Geometry. Asakura Pulishing Co. Ltd. Tokyo (1961).
  • [44] Vaisman, I.: Simplectic Geometry and Secondary Characteristic Classes. Progress in Mathematics. 72, Birkhauser Verlag, Basel (1994).
  • [45] Wong, Y. C.: A global formulation of the condition for a curve to lie in a sphere. Monatschefte fur Mathematik. 67, 363–365 (1963).
  • [46] Wong, Y. C.: On an explicit characterization of spherical curves. Proceedings of the American Math. Soc. 34 (1), 239–242 (1972).

Year 2023, Volume: 16 Issue: 2, 644 - 652, 29.10.2023

Zehra İşbilir , Murat Tosun

https://doi.org/10.36890/iejg.1291893
Cited By: 3

Abstract

Project Number

-

References

  • [1] Akyiğit, M., Yıldız, Ö. G.: On the framed normal curves in Euclidean 4-space. Fundam. J. Math. Appl. 4 (4), 258–263 (2021).
  • [2] Breuer, S., Gottlieb, D.: Explicit characterization of spherical curves. Proc. Amer. Math. Soc. 27 (1), 126–127 (1971).
  • [3] Chen, B.-Y.: When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Monthly. 110 (2), 147–152 (2003).
  • [4] Chen, B.-Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48 (2), 209-–214 (2017).
  • [5] Chen, B.-Y., Dillen, F.: Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Academia Sinica. 33 (2), 77–90 (2005).
  • [6] Constantinescu, O.: Myller configurations in Finsler spaces. Differential Geometry-Dynamical Systems. 8, 69–76 (2006).
  • [7] Deshmukh, S., Chen, B.-Y., Alshammari, S. H.: On rectifying curves in Euclidean 3-space. Turk. J. Math. 42 (2), 609-–620 (2018).
  • [8] Doğan Yazıcı, B., Özkaldı Karaku¸s, S., Tosun, M.: Characterizations of framed curves in four-dimensional Euclidean space. Univers. J. Math. Appl. 4 (4), 125–131 (2021).
  • [9] Gökçelik, F., Bozkurt, Z., Gök, İ, Ekmekçi, F. N., Yaylı, Y.: Parallel transport frame in 4-dimensional Euclidean space. Caspian J. Math. Sci. 3 (1), 91–103 (2014).
  • [10] Gluck, H.: Higher curvatures of curves in Euclidean space. Amer. Math. Monthly. 73, 699–704 (1966).
  • [11] Heroiu, B.: Versor fields along a curve in a four dimensional Lorentz space. J. Adv. Math. Stud. 4 (1), 49–57 (2011).
  • [12] İlarslan, K.: Spacelike normal curves in Minkowski space E31 . Turk. J. Math. 29 (2), 53–63 (2005).
  • [13] İlarslan, K., Nešovic, E.: Spacelike and timelike normal curves in Minkowski space-time. Publications de l’Institut Mathmatique. 85 (99), 111–118 (2009).
  • [14] İlarslan, K., Nešovic, E.: Some characterizations of rectifying curves in Euclidean 4-spaces E4. Turk. J. Math. 32, 21–30 (2008).
  • [15] İlarslan, K., Nešovic, E.: Some characterizations of osculating curves in the Euclidean spaces. Demonstratio Mathematica. 41 (4), 931–940 (2008).
  • [16] İlarslan, K., Nešovic, E.: The first kind and the second kind osculating curves in Minkowski space-time. Compt. Rend. Acad. Bulg. Sci. 62 (6), 677–686 (2009).
  • [17] İlarslan, K., Nešovic, E., Petrovic–Torgašev, M.: Some characterizations of rectifying curves in the Minkowski 3–space. Novi Sad J. Math.. 33 (2), 23-–32 (2003).
  • [18] İlarslan, K., Nešovic, E.: Timelike and null normal curves in Minkowski space E31 . Indian J. Pure Appl. Math. 35 (7), 881–888 (2004).
  • [19]İlarslan, K., Nešovic, E.: Some characterizations of null, pseudo null and partially null rectifying curves in Minkowski space-time. Taiwan. J. Math. 12 (5), 1035–1044 (2008).
  • 20] İlarslan, K., Nešovic, E.: Spacelike and timelike normal curves in Minkowski space-time. Publications de l’Institut Mathématique. 85 (99), 111–118 (2009).
  • [21] İlarslan, K., Nešovi´c, E.: Some characterizations of pseudo and partially null osculating curves in Minkowski space-time. Int. Electron. J. Geom. 4 (2), 1–12 (2011).
  • [22] İlarslan, K., Nešovic, E.: On rectifying curves as centrodes and extremal curves in the Minkowski 3-space. Novi Sad J. Math.. 37 (1), 53–64 (2007).
  • [23] İşbilir, Z., Tosun, M.: On generalized osculating-type curves in Myller configuration. An. Şt. Univ. Ovidius Constanta, Ser. Mat. XXXII (2), (2024).
  • [24] İşbilir, Z., Tosun, M.: An extended framework for osculating-type curves in four-dimensional Euclidean space. (2023) (submitted).
  • [25] Keskin, Ö., Yaylı, Y.: Rectifying-type curves and rotation minimizing frame Rn. arXiv preprint, (2019). https://doi.org/10.48550/ arXiv.1905.04540.
  • [26] Kişi, ˙I.: Some characterizations of canal surfaces in the four dimensional Euclidean space, Ph.D. Thesis, Kocaeli University, (2018).
  • [27] Kuhnel, W.: Differential Geometry: Curves-Surfaces-Manifolds, Braunschweig, Wiesbaden (1999).
  • [28] Levi-Civita, T.: Rendiconti del Circolo di Palermo. (XLII), 173 (1917).
  • [29] Levi-Civita, T.: Lezioni di calcolo differentiate assoluto, Zanichelli, (1925).
  • [30] Macsim, G., Mihai, A., Olteanu, A.: Curves in a Myller configuration. International Conference on Applied and Pure Mathematics (ICAPM 2017), Iaşi, November 2-5, 2017.
  • [31] Macsim, G., Mihai, A., Olteanu, A.: On rectifying-type curves in a Myller configuration. Bull. Korean Math. Soc. 56 (2), 383–390 (2019).
  • [32] Macsim, G., Mihai, A., Olteanu, A.: Special curves in a Myller configuration. Proceedings of the 16th Workshop on Mathematics, Computer Science and Technical Education, Department of Mathematics and Computer Science, Volume 2, (2019).
  • [33] Miron, R.: The Geometry of Myller Configurations. Applications to Theory of Surfaces and Nonholonomic Manifolds, Romanian Academy, (2010).
  • [34] Miron, R.: Geometria unor configuratii Myller. Analele ¸St. Univ. VI (3), (1960).
  • [35] Miron, R.: Myller configurations and Vranceanu nonholonomic manifolds. Scientific Studies and Research. 21 (1), (2011).
  • [36] Miron, R.: Configura¸tii Myller M(C, ξi1, Tn−1) în spa¸tii Riemann Vn, Aplica¸tii la studiul hipersuprafe¸telor din Vn. Studii şi Cerc. Şt. Mat. Iaşi. XII (1), (1962).
  • [37] Miron, R.: Configuratii Myller M(C, ξ1 i , Tm) în spa¸tii Riemann Vn. Aplica¸tii la studiul varietˇa¸tilor Vm din Vn. Studii ¸si Cerc. St. Mat. Ia¸si. XIII (1), (1962).
  • [38] Miron, R.: Les configurations de MyllerM(C, ξ1i , Tn−1) dans les espaces de Riemann Vn (I). Tensor. 12 (3), (1962) (Japonia).
  • [39] Miron, R.: Les configurations de MyllerM(C, ξi1, Tm) dans les espaces de Riemann Vn. Tensor. V (1), (1964) (Japonia).
  • [40] Miron, R., Pop, I.: Topologie algebricˇa: omologie, omotopie, spa¸tii de acoperire. Ed. Academiei, (1974).
  • [41] Miron, R., Branzei, D.: Backgrounds of arithmetic and geometry, World Scientific Publishing, S. Pure Math. 23, (1995).
  • [42] Miron, R.: Geometria Configura¸tiilor Myller. Editura Tehnicˇa. Bucure¸sti (1966).
  • [43] Otsuki, T.: Differential Geometry. Asakura Pulishing Co. Ltd. Tokyo (1961).
  • [44] Vaisman, I.: Simplectic Geometry and Secondary Characteristic Classes. Progress in Mathematics. 72, Birkhauser Verlag, Basel (1994).
  • [45] Wong, Y. C.: A global formulation of the condition for a curve to lie in a sphere. Monatschefte fur Mathematik. 67, 363–365 (1963).
  • [46] Wong, Y. C.: On an explicit characterization of spherical curves. Proceedings of the American Math. Soc. 34 (1), 239–242 (1972).
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Zehra İşbilir 0000-0001-5414-5887

Murat Tosun 0000-0002-4888-1412

Project Number -
Early Pub Date October 19, 2023
Publication Date October 29, 2023
Acceptance Date October 15, 2023
Published in Issue Year 2023 Volume: 16 Issue: 2

Cite

APA İşbilir, Z., & Tosun, M. (2023). A New Insight on Rectifying-Type Curves in Euclidean 4-Space. International Electronic Journal of Geometry, 16(2), 644-652. https://doi.org/10.36890/iejg.1291893
AMA İşbilir Z, Tosun M. A New Insight on Rectifying-Type Curves in Euclidean 4-Space. Int. Electron. J. Geom. October 2023;16(2):644-652. doi:10.36890/iejg.1291893
Chicago İşbilir, Zehra, and Murat Tosun. “A New Insight on Rectifying-Type Curves in Euclidean 4-Space”. International Electronic Journal of Geometry 16, no. 2 (October 2023): 644-52. https://doi.org/10.36890/iejg.1291893.
EndNote İşbilir Z, Tosun M (October 1, 2023) A New Insight on Rectifying-Type Curves in Euclidean 4-Space. International Electronic Journal of Geometry 16 2 644–652.
IEEE Z. İşbilir and M. Tosun, “A New Insight on Rectifying-Type Curves in Euclidean 4-Space”, Int. Electron. J. Geom., vol. 16, no. 2, pp. 644–652, 2023, doi: 10.36890/iejg.1291893.
ISNAD İşbilir, Zehra - Tosun, Murat. “A New Insight on Rectifying-Type Curves in Euclidean 4-Space”. International Electronic Journal of Geometry 16/2 (October 2023), 644-652. https://doi.org/10.36890/iejg.1291893.
JAMA İşbilir Z, Tosun M. A New Insight on Rectifying-Type Curves in Euclidean 4-Space. Int. Electron. J. Geom. 2023;16:644–652.
MLA İşbilir, Zehra and Murat Tosun. “A New Insight on Rectifying-Type Curves in Euclidean 4-Space”. International Electronic Journal of Geometry, vol. 16, no. 2, 2023, pp. 644-52, doi:10.36890/iejg.1291893.
Vancouver İşbilir Z, Tosun M. A New Insight on Rectifying-Type Curves in Euclidean 4-Space. Int. Electron. J. Geom. 2023;16(2):644-52.

Cited By

Non-lightlike framed rectifying curves in Minkowski 3-space
Electronic Research Archive
https://doi.org/10.3934/era.2024273
Non-lightlike framed rectifying curves in Minkowski 3-space
Electronic Research Archive
https://doi.org/10.3934/era.2024273.
A New Insight on Rectifying-Type Curves in Euclidean 4-Space
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.1291893
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