W1-Curvature Tensor within the framework of Lorentzian \alpha-Sasakian Manifold
Abstract
The objective of this paper is to investigate the curvature properties of Lorentzian $\alpha$-Sasakian manifolds under specific geometric conditions. In particular, we examine these manifolds when they satisfy the following conditions: $\zeta-\mathcal{W}_1$-flatness, $\varphi-\mathcal{W}_1$-semi-symmetry, and the vanishing of certain curvature operators, specifically $\mathcal{W}_1 \cdot \mathcal{Q} = 0$ and $\mathcal{W}_1 \cdot R = 0$. Through our analysis, we derive several interesting results regarding the geometric structure and behavior of these manifolds under the given conditions.
Full text article
References
S. Tanno, “The automorphism groups of almost contact riemannian manifolds,” Tohoku Math. J., vol. 21, pp. 21–38, 1969. https://doi.org/10.2748/tmj/1178243031.
A. Gray and L. M. Hervella, “The sixteen classes of almost hermitian manifolds and their linear invariants,” Ann. Mat. Pura Appl., vol. 123, no. 4, pp. 35–58, 1980. https://doi.org/10.1007/BF01796539.
S. Dragomir and L. Ornea, Locally conformal K¨ahler geometry, vol. 155 of Progress in Mathematics. Boston, MA: Birkhauser Boston, Inc., 1998.
A. Oubina, “New classes of contact metric structures,” Publ. Math. Debrecen, vol. 32, no. 3–4, pp. 187–193, 1985.
D. E. Blair, Riemannian geometry contact and symplectic manifolds, vol. 203 of Progress in Mathematics. 2002. https://doi.org/10.1007/978-0-8176-4959-3.
J. C. Marrero and D. Chinea, “On trans-sasakian manifolds,” in Proceedings of the XIVth Spanish-Portuguese Conference on Mathematics, (Univ. La Laguna, La Laguna), pp. 655–659, 1990. Puerto de la Cruz, 1989.
D. Janssens and L. Vanhacke, “Almost contact structures and curvature tensors,” Kodai Math. J., vol. 4, pp. 1–27, 1981. https://doi.org/10.2996/kmj/1138036310.
M. M. Tripathi, “Trans-sasakian manifolds are generalized quasi-sasakian,” Nepali Math. Sci. Rep., vol. 18, no. 1–2, pp. 11–14, 1999–2000.
Yildiz and C. Murathan, “On lorentzian α-sasakian manifolds,” Kyungpook Math. J., vol. 45, pp. 95–103, 2005. http://dx.doi.org/10.4236/apm.2012.23024.
D. E. Blair and J. A. Oubina, “Conformal and related changes of metric on the product of two almost contact metric manifolds,” Publications Matematiques, vol. 34, pp. 199–207, 1990.
Rajan, G. P. Singh, P. Prajapati, and A. K. Mishra, “w8-curvature tensor in Lorentzian α-sasakian manifold,” Turk. J. Comput. Math. Educ., vol. 11, no. 3, pp. 1061–1072, 2020.
J. C. Marrero, “The local structure of trans-sasakian manifolds,” Ann. Mat. Pura Appl., vol. 162, no. 4, pp. 77–86, 1992. https://doi.org/10.1007/BF01760000.
M. M. Tripathi and P. Gupta, “T-curvature tensor on a semi-riemannian manifold,” Journal of Advanced Mathematical Studies, vol. 4, pp. 117–129, 2011.
K. Kenmotsu, “A class of almost contact riemannian manifold,” Tohoku Math. J., vol. 24, pp. 93–103, 1972. https://doi.org/10.2748/tmj/1178241594.
P. Prajapati, G. P. Singh, S. Dey, and Y. J. Suh, “Bochner flat lorentzian kahler space-time manifold with (m, ρ)-quasi-einstein soliton,” International Journal of Geometric Methods in Modern Physics, pp. 2650063, 19 pages, 2025. https://doi.org/10.1142/S0219887826500635.
A. K. Mishra, P. Prajapati, Rajan, and G. P. Singh, “On m-projective curvature tensor of lorentzian β-kenmotsu manifold,” Bull. Trans. Univ. Brasov Series-III: Maths and Computer Sci., vol. 4, no. 66, pp. 201–214, 2024. https://doi.org/10.31926/but.mif.2024.4.66.2.12.
C. S. Bagewadi and Venkatesha, “Some curvature tensors on a kenmotsu manifolds,” Tensor New Series, vol. 68, no. 2, pp. 140–147, 2007.
G. P. Singh, P. Prajapati, A. K. Mishra, and Rajan, “Generalized b-curvature tensor within the framework of lorentzian β-kenmotsu manifold,” Int. J. Geom. Methods Mod. Phys., vol. 21, no. 2, pp. 2450125, 14 pages, 2024. https://doi.org/10.1142/S0219887824501251.
P. Prajapati, G. P. Singh, Rajan, and A. K. Mishra, “A study of w8-curvature tensor in lorentzian β-kenmotsu manifold,” Jour. Rajs. Acad. Phys. Scie., vol. 22, no. 3–4, pp. 225–236, 2023.
D. E. Blair, Contact manifolds in Riemannian geometry, vol. 509 of Lecture Notes in Mathematics. Berlin New-York: Springer-Verlag, 1976.
S. K. Chaubey and R. H. Ojha, “On the m-projective curvature tensor of a kenmotsu manifold,” Differential Geometry Dynamical Systems, vol. 12, pp. 52–60, 2010.
U. C. De, “On φ-symmetric kenmotsu manifolds,” International Electronic Journal of Geometry, vol. 1, pp. 33–38, 2008.
G. Ingalahalli and C. S. Bagewadi, “A study on conservative c-bochner curvature tensor in k-contact and kenmotsu manifolds admitting semi-symmetric metric connection,” ISRN Geometry, pp. 1–14, 2012. https://doi.org/10.5402/2012/709243.
S. K. Hui and D. Chakraborty, “Infinitesimal cl-transformations on kenmotsu manifolds,” Bangmod Int. J. Math and Comp. Sci., vol. 3, pp. 1–9, 2017.
K. K. Baishya and P. R. Chowdhury, “Kenmotsu manifold with some curvature conditions,” Annales Univ. Sci. Budapest., vol. 59, pp. 55–65, 2016.
H. G. Nagaraja, D. L. Kiran Kumar, and V. S. Prasad, “Ricci solitons on kenmotsu manifolds under d-homothetic deformation,” Khayyam J. Math., vol. 4, pp. 102–109, 2018. https://doi.org/10.22034/kjm.2018.57725.
C. ¨Ozg¨ur, “On weakly symmetric kenmotsu manifolds,” Differential Geometry-Dynamical Systems, vol. 8, pp. 204–209, 2006.
D. G. Prakasha and B. S. Hadimani, “On the conharmonic curvature tensor of kenmotsu manifolds with generalized tanaka-webster connection,” Misolc. Math. Notes, vol. 19, pp. 491–503, 2018. https://doi.org/10.18514/MMN.2018.1596.
A. A. Shaikh and S. K. Hui, “On locally φ-symmetric β-kenmotsu manifolds,” Extracta Mathematicae, vol. 24, pp. 301–316, 2009.
G. P. Singh, A. K. Mishra, P. Prajapati, and Rajan, “M-projective curvature tensor on ϵ-kenmotsu manifolds,” Southeast Asian Bulletin of Mathematics, vol. 49, no. 5, pp. 697–706, 2025. https://doi.org/10.31926/but.mif.2024.4.66.2.12.
B. B. Sinha and A. K. Srivastava, “Curvature on kenmotsu manifold,” Indian J. Pure and Appl. Math., vol. 22, pp. 23–28, 1991.
M. M. Trıpathı and P. Gupta, “On (n (k); ξ)-semi-riemannian manifolds: Semisymmetries,” International Electronic Journal of Geometry, vol. 5, no. 1, pp. 42–77, 2012.
G. P. Singh, A. K. Mishra, Rajan, P. Prajapati, and A. P. Tiwari, “A study on w8-curvature tensor in ϵ-kenmotsu manifolds,” Ganita, vol. 72, no. 2, pp. 115–126, 2022.
G. P. Singh, Rajan, A. K. Mishra, and P. Prajapati, “w8-curvature tensor in generalized sasakian space form,” Ratio Mathematica, vol. 48, pp. 51–62, 2023. https://doi.org/10.23755/rm.v48i0.961.
Authors
Copyright (c) 2025 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
