LP-Kenmotsu Manifolds Admitting Bach Almost Solitons

Yıl 2024, Cilt: 7 Sayı: 3, 102 - 110

Rajendra Prasad , Abhinav Verma , Vindhyachal Singh Yadav , Abdul Haseeb , Mohd Bilal

https://doi.org/10.32323/ujma.1443527

Öz

For a Lorentzian para-Kenmotsu manifold of dimension $m$ (briefly, ${(LPK)_{m}}$) admitting Bach almost soliton $(g,\zeta,\lambda)$, we explored the characteristics of the norm of Ricci operator. Besides, we gave the necessary condition for ${(LPK)_{m}}$ ($m\geq 4$) admitting Bach almost soliton to be an $\eta$-Einstein manifold. Afterwards, we proved that Bach almost solitons are always steady when a Lorentzian para-Kenmotsu manifold of dimension three has Bach almost soliton.

Anahtar Kelimeler

Bach almost solitons, $LP$-Kenmotsu manifolds, Perfect fluid, Weyl tensor

Kaynakça

• [1] I. Sato, On a structure similar to the almost contact structure, Tensor (N. S.), 30 (1976), 219-224.
• [2] S. Kaneyuki, M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math., 8 (1985), 81-98.
• [3] S. Kaneyuki, F. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173-187.
• [4] B.B. Sinha, K.L. Sai Prasad, A class of almost para contact metric manifolds, Bull. Calcutta Math. Soc., 87 (1995), 307-312.
• [5] T. Q. Binh, L. Tamassy, U. C. De, M. Tarafdar, Some remarks on almost Kenmotsu manifolds, Maths. Pannonica, 13 (2002), 31-39.
• [6] A. M. Blaga, Conformal and paracontactly geodesic transformations of almost paracontact metric structures, Facta Univ. Scr. Math. Inform., 35 (2020), 121-130.
• [7] A. M. Blaga, M. Crasmareanu, Statistical structures in almost paracontact geometry, Bull. Iranian Math. Soc., 44 (2018), 1407-1413.
• [8] G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55 (2011), 696-718.
• [9] S. Dirik, M. Atceken, U. Yildirim, Anti invariant submanifolds of a normal para contact metric manifolds, Gulf J. Math., 10 (2021), 38-49.
• [10] B. O’Neill, Semi-Riemannian Geometry with Application to Relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, New York, 1983.
• [11] V. R. Kaigorodov, The curvature structure of spacetime, Pro. Geom., 14 (1983), 177-202.
• [12] A. K. Raychaudhuri, S. Banerji, A. Banerjee, General Relativity, Astrophysics and Cosmology, Springer- Verlag, 1992.
• [13] A. Haseeb, R. Prasad, Certain results on Lorentzian para-Kenmotsu manifolds, Bol. Soc. Paran. Mat., 39 (2021), 201-220.
• [14] A. Haseeb, R. Prasad, Some results on Lorentzian para-Kenmotsu manifolds, Bulletin of the Transilvania University of Brasov, Series III : Mathematics, Informatics, Physics, 13(62) (2020), 185-198.
• [15] R. Bach, Zur Weylschen relativitatstheorie and der Weylschen Erweiterung des Krummungstensorbegriffs, Math. Z., 9 (1921), 110-135.
• [16] Y. Wang, Cotton tensors on almost co-K¨ahler 3-manifolds, Ann. Polon. Math., 120 (2017), 135-148.
• [17] J. Bergman, Conformal Einstein spaces and Bach tensor generalization in n-dimensions, Ph. D. Thesis, Link¨oping University, 2004.
• [18] U. C. De, K. De, On a class of three-dimensional trans-Sasakian manifolds, Commun. Korean Math. Soc., 27 (2012), 795-808.
• [19] U. C. De, G. Ghosh, J. B. Jun, P. Majhi, Some results on para Sasakian manifolds, Bull. Transilv. Univ. Brasov, Series III : Mathematics, Informatics, Physics, 11 (60) (2018), 49-63.
• [20] U. C. De, A. Sardar, Classification of (k;m)-almost co-K¨ahler manifolds with vanishing Bach tensor and divergence free cotton tensor, Commun. Korean Math. Soc., 35 (2020), 1245-1254.
• [21] H.I. Yoldas, Notes on h-Einstein solitons on para-Kenmotsu manifolds, Math. Method Appl. Sci., 46 (2023), 17632-17640.
• [22] H. Fu, J. Peng, Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature, Hokkaido Math. J., 47 (2018), 581-605.
• [23] A. Ghosh, R. Sharma, Sasakian manifolds with purely transversal Bach tensor, J. Math. Phys., 58 (2017), 103502, 6 pp.
• [24] P. Szekeres, Conformal tensors, Proc. R. Soc. Lond. Ser. A-Contain. Pap. Math. Phys., 304 (1968), 113-122.
• [25] S. Das, S. Kar, Bach flows of product manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250039, 18 pp.
• [26] I. Bakas, F. Bourliot, D. Lust, M. Petropoulos, Geometric flows in Horava-Lifshitz gravity, J. High Energy Phys., 2010 (2010), 1-58.
• [27] E. Bahuaud, D. Helliwell, Short-time existence for some higher-order geometric flows, Commun. Partial Differ. Equ., 36 (2011), 2189-2207.
• [28] H.-D. Cao, Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J., 162 (2013), 1149-1169.
• [29] P. T. Ho, Bach flow, J. Geom. Phys., 133 (2018), 1-9.
• [30] D. Helliwell, Bach flow on homogeneous products, SIGMA Symmetry Integrability Geom. Methods Appl., 16 (2020), 35 pp.
• [31] A. Ghosh, On Bach almost solitons, Beitr. Algebra Geom., 63 (2022), 45-54.
• [32] K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Sci., 12 (1989), 151-156.
• [33] R. Prasad, A. Haseeb, A. Verma, V. S. Yadav, A study of j-Ricci symmetric LP-Kenmotsu manifolds, Int. J. Maps Math., 7 (2024), 33-44.
• [34] Pankaj, S. K. Chaubey, R. Prasad, Three dimensional Lorentzian para-Kenmotsu manifolds and Yamabe soliton, Honam Math. J., 43 (2021), 613-626.
• [35] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Springer-Verlag, 509, 1976.
• [36] Li, Y., Haseeb, A., Ali, M., LP-Kenmotsu manifolds admitting h-Ricci solitons and spacetime, J. Math., 2022, Article ID 6605127, 10 pages.