Communications of the Korean Mathematical Society (대한수학회논문집)

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YAMABE AND RIEMANN SOLITONS ON LORENTZIAN PARA-SASAKIAN MANIFOLDS

  • Received : 2020.09.24
  • Accepted : 2021.09.07
  • Published : 2022.01.31
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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

Acknowledgement

The first author (Shruthi Chidananda) is thankful to University Grants Commission, New Delhi, India (Ref. No.: 1019/(ST)(CSIR-UGC NET DEC. 2016) for financial support in the form of UGC-Junior Research Fellowship. The author also thankful to DST, New Delhi, for providing financial assistance under FIST programme.

References

  1. 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
  2. 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
  3. 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.
  4. S. Chidananda and V. Venkatesha, Riemann soliton on non-Sasakian (κ, µ)-contact manifolds, Differ. Geom. Dyn. Syst. 23 (2021), 40-51.
  5. 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
  6. K. De and U. C. De, A note on almost Riemann soliton and gradient almost Riemann soliton, https://arxiv.org/abs/2008.10190.
  7. S. Deshmukh and B. Y. Chen, A note on Yamabe solitons, Balkan J. Geom. Appl. 23 (2018), no. 1, 37-43.
  8. 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
  9. 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
  10. 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
  11. 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
  12. I. E. Hirica and C. Udriste, Ricci and Riemann solitons, Balkan J. Geom. Appl. 21 (2016), no. 2, 35-44.
  13. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Natur. Sci. 12 (1989), no. 2, 151-156.
  14. K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor (N.S.) 47 (1988), no. 2, 189-197.
  15. 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.
  16. I. Mihai, A. A. Shaikh, and U. C. De, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Matematico di Messina, Serie II. (1999) 3.
  17. 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
  18. 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. https://doi.org/10.15407/mag16.04.402
  19. 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.
  20. M. Tarafdar and A. Bhattacharyya, On Lorentzian para-Sasakian manifolds, in Steps in differential geometry (Debrecen, 2000), 343-348, Inst. Math. Inform., Debrecen, 2001.
  21. C. Udriste, Riemann flow and Riemann wave, An. Univ. Vest Timi,s. Ser. Mat.-Inform. 48 (2010), no. 1-2, 265-274.
  22. C. Udriste, Riemann flow and Riemann wave via bialternate product Riemannian metric, preprint (2012). arXiv.org/math.DG/1112.4279v4
  23. 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
  24. 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
  25. 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 https://doi.org/10.36045/bbms/1473186509
  26. 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
  27. 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
  28. K. Yano, Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.