Browse > Article
http://dx.doi.org/10.4134/CKMS.c200365

YAMABE AND RIEMANN SOLITONS ON LORENTZIAN PARA-SASAKIAN MANIFOLDS  

Chidananda, Shruthi (Department of Mathematics Kuvempu University)
Venkatesha, Venkatesha (Department of Mathematics Kuvempu University)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.1, 2022 , pp. 213-228 More about this Journal
Abstract
In the present paper, we aim to study Yamabe soliton and Riemann soliton on Lorentzian para-Sasakian manifold. First, we proved, if the scalar curvature of an 𝜂-Einstein Lorentzian para-Sasakian manifold M is constant, then either 𝜏 = n(n-1) or, 𝜏 = n-1. Also we constructed an example to justify this. Next, it is proved that, if a three dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton for V is an infinitesimal contact transformation and tr 𝜑 is constant, then the soliton is expanding. Also we proved that, suppose a 3-dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton, if tr 𝜑 is constant and scalar curvature 𝜏 is harmonic (i.e., ∆𝜏 = 0), then the soliton constant λ is always greater than zero with either 𝜏 = 2, or 𝜏 = 6, or λ = 6. Finally, we proved that, if an 𝜂-Einstein Lorentzian para-Sasakian manifold M represents a Riemann soliton for the potential vector field V has constant divergence then either, M is of constant curvature 1 or, V is a strict infinitesimal contact transformation.
Keywords
Lorentzian para-Sasakian manifold; ${\eta}$-Einstein manifold; Yamabe soliton; Riemann soliton;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 Y. Wang, Yamabe solitons on three-dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), no. 3, 345-355. http://projecteuclid.org/euclid.bbms/1473186509   DOI
2 V. Venkatesha, H. A. Kumara, and D. M. Naik, Riemann solitons and almost Riemann solitons on almost Kenmotsu manifolds, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 7, 2050105, 22 pp. https://doi.org/10.1142/S0219887820501054   DOI
3 Y. Wang, Minimal and harmonic Reeb vector fields on trans-Sasakian 3-manifolds, J. Korean Math. Soc. 55 (2018), no. 6, 1321-1336. https://doi.org/10.4134/JKMS.j170689   DOI
4 K. De and U. C. De, A note on almost Riemann soliton and gradient almost Riemann soliton, https://arxiv.org/abs/2008.10190.
5 S. Deshmukh and B. Y. Chen, A note on Yamabe solitons, Balkan J. Geom. Appl. 23 (2018), no. 1, 37-43.
6 K. Yano, Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.
7 A. M. Blaga and D. R. Latcu, Remarks on Riemann and Ricci solitons in (α, β)-contact metric manifolds, J. Geom. Symmetry Phys. 58 (2020), 1-12. https://doi.org/10.7546/jgsp-58-2020-1-12   DOI
8 C. Udriste, Riemann flow and Riemann wave, An. Univ. Vest Timi,s. Ser. Mat.-Inform. 48 (2010), no. 1-2, 265-274.
9 S. K. Chaubey, Some properties of LP-Sasakian manifolds equipped with m-projective curvature tensor, Bull. Math. Anal. Appl. 3 (2011), no. 4, 50-58.
10 B.-Y. Chen and S. Deshmukh, Yamabe and quasi-Yamabe solitons on Euclidean sub-manifolds, Mediterr. J. Math. 15 (2018), no. 5, Paper No. 194, 9 pp. https://doi.org/10.1007/s00009-018-1237-2   DOI
11 M. N. Devaraja, H. Aruna Kumara, and V. Venkatesha, Riemann soliton within the framework of contact geometry, Quaest. Math. 44 (2021), no. 5, 637-651. https://doi.org/10.2989/16073606.2020.1732495   DOI
12 K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Natur. Sci. 12 (1989), no. 2, 151-156.
13 K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor (N.S.) 47 (1988), no. 2, 189-197.
14 A. M. Blaga, Some geometrical aspects of Einstein, Ricci and Yamabe solitons, J. Geom. Symmetry Phys. 52 (2019), 17-26. https://doi.org/10.7546/jgsp-52-2019-17-26   DOI
15 M. Tarafdar and A. Bhattacharyya, On Lorentzian para-Sasakian manifolds, in Steps in differential geometry (Debrecen, 2000), 343-348, Inst. Math. Inform., Debrecen, 2001.
16 V. Venkatesha and D. M. Naik, Yamabe solitons on 3-dimensional contact metric manifolds with Qϕ = ϕQ, Int. J. Geom. Methods Mod. Phys. 16 (2019), no. 3, 1950039, 9 pp. https://doi.org/10.1142/S0219887819500397   DOI
17 I. Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, in Classical analysis (Kazimierz Dolny, 1991), 155-169, World Sci. Publ., River Edge, NJ, 1992.
18 I. Mihai, A. A. Shaikh, and U. C. De, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Matematico di Messina, Serie II. (1999) 3.
19 D. M. Naik, V. Venkatesha, and H. A. Kumara, Ricci solitons and certain related metrics on almost co-Kaehler manifolds, Zh. Mat. Fiz. Anal. Geom. 16 (2020), no. 4, 402-417.   DOI
20 A. A. Shaikh and K. K. Baishya, Some results on LP-Sasakian manifolds, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 49(97) (2006), no. 2, 193-205.
21 C. Udriste, Riemann flow and Riemann wave via bialternate product Riemannian metric, preprint (2012). arXiv.org/math.DG/1112.4279v4
22 D. M. Naik, Ricci solitons on Riemannian manifolds admitting certain vector field, Ricerche di Matematica (2021). https://doi.org/10.1007/s11587-021-00622-z   DOI
23 S. Chidananda and V. Venkatesha, Riemann soliton on non-Sasakian (κ, µ)-contact manifolds, Differ. Geom. Dyn. Syst. 23 (2021), 40-51.
24 Y. Wang, Almost Kenmotsu (k, µ)'-manifolds with Yamabe solitons, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115 (2021), no. 1, Paper No. 14, 8 pp. https://doi.org/10.1007/s13398-020-00951-y   DOI
25 A. Ghosh, Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold, Math. Slovaca 70 (2020), no. 1, 151-160. https://doi.org/10.1515/ms-2017-0340   DOI
26 R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. https://doi.org/10.1090/conm/071/954419   DOI
27 I. E. Hirica and C. Udriste, Ricci and Riemann solitons, Balkan J. Geom. Appl. 21 (2016), no. 2, 35-44.
28 I. K. Erken, Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Period. Math. Hungar. 80 (2020), no. 2, 172-184. https://doi.org/10.1007/s10998-019-00303-3   DOI