SavedAdd your journal

Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 2

Canal Hypersurfaces Generated by Pseudo Null, Partially Null and Null Curves in Lorentz-Minkowski 4-Space

Mustafa Altın(BINGOL UNIVERSITY)0000-0001-5544-5910
Ahmet Kazan(MALATYA TURGUT ÖZAL ÜNİVERSİTESİ)0000-0002-1959-6102
Dae Won Yoon(Gyeongsang National University)0000-0001-8620-0676
Pages
—
Publication Date
2026-10-25
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Also, we construct some examples for these canal hypersurfaces and finally, we give some characterizations for tubular hypersurfaces in $E_{1}^{4}$.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we obtain the parametric expressions of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on a pseudo null, partially null or null curve in $E_{1}^{4}$ and give some geometric invariants such as unit normal vector fields, Gauss-Kronecker curvatures and mean curvatures of the ones generated by pseudo null and partially null curves.
📊
Key Findings & Data:In this paper, we obtain the parametric expressions of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on a pseudo null, partially null or null curve in $E_{1}^{4}$ and give some geometric invariants such as unit normal vector fields, Gauss-Kronecker curvatures and mean curvatures of the ones generated by pseudo null and partially null curves.
💡
Conclusion & Impact:Also, we construct some examples for these canal hypersurfaces and finally, we give some characterizations for tubular hypersurfaces in $E_{1}^{4}$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Pseudo null curve · partially null curve · null curve

Abstract

In this paper, we obtain the parametric expressions of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres or pseudo hyperbolic hyperspheres whose centers lie on a pseudo null, partially null or null curve in $E_{1}^{4}$ and give some geometric invariants such as unit normal vector fields, Gauss-Kronecker curvatures and mean curvatures of the ones generated by pseudo null and partially null curves. Also, we construct some examples for these canal hypersurfaces and finally, we give some characterizations for tubular hypersurfaces in $E_{1}^{4}$.

🔖

Cite This Article

Altın, M., Kazan, A., Yoon, D. W. (2026). Canal Hypersurfaces Generated by Pseudo Null, Partially Null and Null Curves in Lorentz-Minkowski 4-Space. International Electronic Journal of Geometry, Volume: 19 Issue: 2. https://doi.org/10.36890/iejg.1920156

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (43)

  1. Aksoyak, F. K., Gök, İ., İlar­slan, K.: Generalized Null Bertrand Curves in Minkowski Space-Time. Ann. Alexandru Ioan Cuza Univ. Math. 60, 489–502 (2014). DOI:10.2478/auom-2013-0031
  2. Akyiğit, M., Eren, K., Kosal, H. H.: Tubular Surfaces with Modified Orthogonal Frame in Euclidean 3-Space. Honam Math. J. 43 (3), 453–463 (2021). DOI:10.5831/HMJ.2021.43.3.453
  3. Altın, M.: Rotational hypersurfaces generated by weighted mean curvature in $E_1^4$ with density. Thermal Science 26 (4A), 3043–3053 (2022). https://doi.org/10.2298/TSCI2204043A
  4. Altın, M., Kazan, A., Karadağ, H. B.: Monge Hypersurfaces in Euclidean 4-Space with Density. Journal of Polytechnics 23 (1), 207–214 (2020). https://doi.org/10.2339/politeknik.634175
  5. Altın, M., Kazan, A., Yoon, D. W.: 2-Ruled Hypersurfaces in Euclidean 4-space. Journal of Geometry and Physics 166, 104236 (2021). https://doi.org/10.1016/j.geomphys.2021.104236
  6. Altın, M., Kazan, A.: Rotational Hypersurfaces in Lorentz-Minkowski 4-Space. Hacettepe Journal of Mathematics and Statistics 50 (5), 1409–1433 (2021). https://doi.org/10.15672/hujms.826596
  7. Altın, M., Kazan, A., Yoon, D. W.: Canal Hypersurfaces generated by Non-Null Curves in Lorentz-Minkowski 4-Space. Bull. Korean Math. Soc. 60 (5), 1299–1320 (2023). https://doi.org/10.4134/BKMS.b220680
  8. Aslan, S., Yaylı, Y.: Canal Surfaces with Quaternions. Adv. Appl. Clifford Algebr. 26 (1), 31–38 (2016). https://doi.org/10.1007/s00006-015-0602-5
  9. Aydın, M. E., Mihai, I.: On certain surfaces in the isotropic 4-space. Mathematical Communications 22 (1), 41–51 (2017).
  10. Bonnor, W. B.: Null curves in a Minkowski space-time. Tensor 20 (2), 229–242 (1969).
  11. Bonnor, W. B.: Curves with null normals in Minkowski space-time. A random walk in relativity and cosmology, Wiley Eastern Limited, 33–47 (1985).
  12. Chen, B. Y.: Pseudo-Riemannian Geometry, $\delta$-invariants and Applications. World Scientific (2011).
  13. Doğan, E., Yaylı, Y.: Tubes with Darboux Frame. Int. J. Contemp. Math. Sci. 7 (16), 751–758 (2012).
  14. Duggal, K. L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic Publishers, The Netherlands (1996).
  15. Duggal, K. L., Jin, D. H.: Null Curves and Hypersurfaces of Semi-Riemannian Manifolds. World Scientific, Singapore (2007).
  16. Duggal, K. L., Şahin, B.: Differential Geometry of Lightlike Submanifolds. Basel: Birkhäuser (2010).
  17. Fu, X., Jung, S. D., Qian, J., Su, M.: Geometric Characterizations of Canal Surfaces in Minkowski 3-space. Bull. Korean Math. Soc. 56 (4), 867–883 (2019). https://doi.org/10.4134/BKMS.b180643
  18. Garcia, R., Libre, J., Sotomayor, J.: Lines of Principal Curvature on Canal Surfaces in $R^3$. An. Acad. Brasil. Cienc. 78 (3), 405–415 (2006). DOI:10.1590/S0001-37652006000300002
  19. Güler, E.: Helical Hypersurfaces in Minkowski Geometry $E_1^4$. Symmetry 12 (8), 1206 (2020). https://doi.org/10.3390/sym12081206
  20. Hartmann, E.: Geometry and Algorithms for Computer Aided Design. Dept. of Math. Darmstadt Univ. of Technology 95 (2003).
  21. İzumiya, S., Takeuchi, M.: On caustics of submanifolds and canal hypersurfaces in Euclidean space. Topology and its Applications 159 (2), 501–508 (2012). https://doi.org/10.1016/j.topol.2011.09.025
  22. İlarslan, K., Nešović, E.: Some Characterizations of Null, Pseudo Null and Partially Null Rectifying Curves in Minkowski Space-Time. Taiwanese Journal of Mathematics 12 (5), 1035–1044 (2008). https://doi.org/10.11650/twjm/1500547425
  23. İlarslan, K., Nešović, E.: Some Characterizations of Pseudo and Partially Null Osculating Curves in Minkowski Space-Time. International Electronic Journal of Geometry 4 (2), 1–12 (2011). https://izik.org/JA4U3F4A
  24. Karacan, M. K., Es, H., Yaylı, Y.: Singular Points of Tubular Surfaces in Minkowski 3-Space. Sarajevo J. Math. 2 (1), 73–82 (2006). https://doi.org/10.5644/SJM.02.1.08
  25. Karacan, M. K., Tunçer, Y.: Tubular Surfaces of Weingarten Types in Galilean and Pseudo-Galilean. Bull. Math. Anal. Appl. 5 (2), 87–100 (2013).
  26. Karacan, M. K., Yoon, D. W., Tunçer, Y.: Tubular Surfaces of Weingarten Types in Minkowski 3-Space. Gen. Math. Notes 22 (1), 44–56 (2014).
  27. Karacan, M. K., Bükcü, B.: An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space. Rendiconti del Circolo Matematico di Palermo 57 (2), 193–201 (2008). https://doi.org/10.1007/s12215-008-0013-8
  28. Kazan, A., Altın, M., Yoon, D. W.: Geometric Characterizations of Canal Hypersurfaces in Euclidean Spaces. Filomat 37 (18), 5909–5920 (2023). https://doi.org/10.2298/FIL23180909K
  29. Kazan, A.: Canal Hypersurfaces according to one of the Extended Darboux Frame Field in Euclidean 4-Space. Thermal Science 26 (4A), 3029–3041 (2022). https://doi.org/10.2298/TSCI2204029K
  30. Kızıl­tug, S., Dede, M., Ekici, C.: Tubular Surfaces with Darboux Frame in Galilean 3-Space. Facta Universitatis Ser. Math. Inform. 34 (2), 253–260 (2019). DOI:10.22190/FUMI1902253K
  31. Kim, Y. H., Liu, H., Qian, J.: Some Characterizations of Canal Surfaces. Bull. Korean Math. Soc. 53 (2), 461–477 (2016). https://doi.org/10.4134/BKMS.2016.53.2.461
  32. Kişi, İ., Öztürk, G., Arslan, K.: A new type of canal surface in Euclidean 4-space $E^4$. Sakarya University Journal of Science 23 (5), 801–809 (2019). https://doi.org/10.16984/saufenbilder.524471
  33. Krivoshapko, S. N., Hyeng, C. A. B.: Classification of Cyclic Surfaces and Geometrical Research of Canal Surfaces. International Journal of Research and Reviews in Applied Sciences 12 (3), 360–374 (2012).
  34. Kühnel, W.: Differential Geometry: Curves-Surfaces-Manifolds. Vieweg, Braunschweig/Wiesbaden (1999).
  35. Lee, J. M.: Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag, New York, Inc. (1997).
  36. Maekawa, T., Patrikalakis, N. M., Sakkalis, T., Yu, G.: Analysis and Applications of Pipe Surfaces. Comput. Aided Geom. Design 15 (5), 437–458 (1998). https://doi.org/10.1016/S0167-8396(97)00042-3
  37. O'Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, New York (1983).
  38. Peternell, M., Pottmann, H.: Commuting Rational Parametrizations of Canal Surfaces. J. Symbolic Comput. 23 (2–3), 255–266 (1997).
  39. Roj, J. S., Yoon, D. W.: Tubes of Weingarten Type in a Euclidean 3-Space. Journal of the Chungcheong Mathematical Society 22 (3), 359–366 (2009).
  40. Uçum, A., İlarslan, K.: New Types of Canal Surfaces in Minkowski 3-Space. Adv. Appl. Clifford Algebr. 26, 449–468 (2016). https://doi.org/10.1007/s00006-015-0556-7
  41. Walrave, J.: Curves and Surfaces in Minkowski Space. Doctoral thesis, KU Leuven, Faculty of Science, Leuven (1995).
  42. Xu, Z., Feng, R., Sun, J.-G.: Analytic and Algebraic Properties of Canal Surfaces. J. Comput. Appl. Math. 195, 220–228 (2006). https://doi.org/10.1016/j.cam.2005.08.002
  43. Yoon, D. W., Yüzbaşı, Z. K.: Tubular Surfaces with Galilean Darboux Frame in $G_3$. Journal of Mathematical Physics, Analysis, Geometry 15 (2), 278–287 (2019). https://doi.org/10.15407/mag15.02.278