SavedAdd your journal

Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 1

Geometric Characterizations of Generalized $\mathcal{Z}$-solitons on Lorentzian Trans-Sasakian $3$-manifolds

Semra Zeren(SIVAS SCIENCE AND TECHNOLOGY UNIVERSITY)0000-0003-0083-3507
Ahmet Yıldız(INONU UNIVERSITY)0000-0002-9799-1781
Pages
—
Publication Date
2026-04-22
🏛️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)

We also study the existence of generalized $\mathcal{Z}$-Ricci solitons and prove that the existence of such a soliton implies that the scalar curvature is constant and the soliton generally exhibits a shrinking behavior.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we investigate the geometric characterizations of $3$-dimensional Lorentzian trans-Sasakian manifolds admitting a generalized $\mathcal{Z}$-tensor.
📊
Key Findings & Data:First, we analyze the behavior of the generalized $\mathcal{Z}$-tensor under certain symmetry conditions, such as Codazzi type, cyclic parallel, and $\phi$-$\mathcal{Z}$ symmetric conditions, and derive the necessary constraints on the manifold structure.
💡
Conclusion & Impact:We also study the existence of generalized $\mathcal{Z}$-Ricci solitons and prove that the existence of such a soliton implies that the scalar curvature is constant and the soliton generally exhibits a shrinking behavior.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Lorentzian trans-Sasakian manifold · generalized $\mathcal{Z}$-tensor · generalized $\mathcal{Z}$-soliton

Abstract

In this paper, we investigate the geometric characterizations of $3$-dimensional Lorentzian trans-Sasakian manifolds admitting a generalized $\mathcal{Z}$-tensor. First, we analyze the behavior of the generalized $\mathcal{Z}$-tensor under certain symmetry conditions, such as Codazzi type, cyclic parallel, and $\phi$-$\mathcal{Z}$ symmetric conditions, and derive the necessary constraints on the manifold structure. We also study the existence of generalized $\mathcal{Z}$-Ricci solitons and prove that the existence of such a soliton implies that the scalar curvature is constant and the soliton generally exhibits a shrinking behavior.

🔖

Cite This Article

Zeren, S., Yıldız, A. (2026). Geometric Characterizations of Generalized $\mathcal{Z}$-solitons on Lorentzian Trans-Sasakian $3$-manifolds. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1881899

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (22)

  1. Azami, S., De, U.C.: Generalized Z-Solitons on Magneto-Fluid Spacetimes in f(r)-Gravity. International Journal of Theoretical Physics. 64 (33), (2025). https://doi.org/10.1007/s10773-025-05900-2
  2. Barman, A., Ünal, ˙I.: Geometry of Kenmotsu manifolds admitting Z-tensor. Bulletin of the Transilvania University of Bra¸sov, Series III: Mathematics and Computer Science. 2 (64), 23-40 (2022). https://doi.org/10.31926/but.mif.2022.2.64.1.3
  3. Blair, D. E.: Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics. Vol 509, Springer-Verlag, Berlin-New York (1976). https://doi.org/10.1007/BFb0079307
  4. De, U.C., De, K.: On Lorentzian trans-Sasakian manifolds. Commun. Fac. Sci. Univ. Ank. series A1 Mathematics and Statistics. 62 (2), 37-51 (2013). https://doi.org/10.1501/Commua1_0000000697
  5. De, U.C., De, K., Shenawy, S.: Characterizations of Pseudo Z-symmetric twisted spacetimes. Indian J Phys. (2026). https://doi.org/10.1007/s12648-025-03910-8
  6. De, K., De, U.C.: Investigation of generalized Z-recurrent spacetimes and f(R, T)- gravity. Advances in Applied Clifford Algebras. 31(3) (2021). DOI: 10.1007/s00006-021-01145-4
  7. De, K., De, U.C.: Certain applications of Pseudo generalized Z-recurrent spacetimes in modified gravity. Modern Physics Letters A. 40 (36), 2550179 (2025). DOI: 10.1142/S0217732325501792
  8. Hamilton, R.S.: The Ricci flow on surfaces. Mathematics and General Relativity. Contemporary Mathematics. 71, 237-262 (1988). https://doi.org/10.1090/conm/071/954419
  9. Mantica, C.A., Molinari, L.G.: Weakly Z-symmetric manifolds. Acta Mathematica Hungarica. 135 (1–2), 80–96 (2012). DOI: 10.1007/s10474-011-0166-3
  10. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York, (1983).
  11. Oubina, J.A.: New classes of almost contact metric structures. Publicationes Mathematicae Debrecen. 32 (4), 187-193 (1985).
  12. Pandey, P.: On Weakly Cyclic Generalized Z-Symmetric Manifolds. National Academy Science Letters. 43 (1), 347-350 (2020). DOI: 10.1007/s40009-020-00875-6
  13. Pandey, P., Sharma, K.: Generalized Z-Ricci soliton on Sasakian 3- manifold with generalized Z-tensor. In: AIP Conference Proceedings, 3087, 060001, (2024). DOI: 1https://doi.org/10.1063/5.0199390
  14. Siddiqi, M.D., Akyol, M.A.: η-Ricci solitons and gradient Ricci solitons on δ-Lorentzian trans-Sasakian manifolds. Facta Universitatis, Series: Mathematics and Informatics. 36 (3) 529-545 (2021). https://doi.org/10.22190/FUMI201010039S
  15. Siddiqi, M.D., Chaubey, S.K., Khan, M.N.I.: f(R, T)- gravity model with perfect fluid admitting Einstein solitons. Mathematics. 10 (1), 82 (2022). https://doi.org/10.3390/math10010082
  16. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field equations, Second Edition. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511535185
  17. Suh, Y.J., De, U.C.: On a type of spacetimes. Filomat. 33 (13), 4251–4260 (2019). https://doi.org/10.2298/FIL1913251S
  18. Yildirim, M., Akyol, M. A., Choudhary, M. A., Aloui, F.: Z-solitons and gradient Z-solitons on α- cosymplectic manifolds. Axioms. 14, 759 (2025). https://doi.org/10.3390/axioms14100759
  19. Yildiz, A., Turan, M., Murathan, C.: A class of Lorentzian α−Sasakian manifolds. Kyungpook Math. J. 49 (4), 789- 799 (2009). https://doi.org/10.5666/kmj.2009.49.4.789
  20. Zeren, S., Yildiz, A.: On Trans- Sasakian Manifolds with the Schouten- van Kampen Connection. Konuralp Journal of Mathematics. 8 (1), 152-157 (2020).
  21. Zeren, S., Yildiz, A.: On trans- Sasakian 3-manifolds with respect to the Schouten- van Kampen connection. International Journal of Maps in Mathematics. 4 (2), 107-120 (2021).
  22. Zeren, S., Yıldız, A., Yüksel, P.S.:On trans- Sasakian 3-manifolds with Da- homotetic deformation with regard to the Schouten- van Kampen connection. Publications de l’Institut Mathematique. 112 (126), 71-82 (2022). DOI: 10.2298/PIM2226071Z