Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

𝜂-RICCI SOLITONS ON 𝜖 - LP-SASAKIAN MANIFOLDS WITH A QUARTER-SYMMETRIC METRIC CONNECTION

  • Haseeb, Abdul (Department of Mathematics, Faculty of Science, Jazan University) ;
  • Prasad, Rajendra (Department of Mathematics and Astronomy, University of Lucknow)
  • Received : 2018.12.05
  • Accepted : 2019.02.23
  • Published : 2019.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study ${\eta}$-Ricci solitons on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions. In particular, we have discussed that the Ricci soliton on ${\epsilon}$-LP-Sasakian manifolds with a quarter-symmetric metric connection satisfying certain curvature conditions is expanding or steady according to the vector field ${\xi}$ being timelike or spacelike. Moreover, we construct 3-dimensional examples of an ${\epsilon}$-LP-Sasakian manifold with a quarter-symmetric metric connection to verify some results of the paper.

Keywords

References

  1. A. Bejancu and K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, Int. J. Math. Math. Sci., 16 (1993), 545-556. https://doi.org/10.1155/S0161171293000675
  2. A. Haseeb, Some results on projective curvature tensor in an ${\epsilon}$-Kenmotsu manifold, Palestine J. Math., 6(Special Issue: II) (2017), 196-203.
  3. A. Haseeb, A. Prakash and M. D. Siddiqi, On a quarter-symmetric metric connec- tion in an ${\epsilon}$-Lorentzian para-Sasakian manifold, Acta Math. Univ. Comenianae, 86(1) (2017), 143-152.
  4. A. Haseeb, M. A. Khan. and M. D. Siddiqi, Some more results on an ${\epsilon}$-Kenmotsu manifold with a semi-symmetric metric connection, Acta Math. Univ. Comenianae, 85(1) (2016), 9-20.
  5. A. Singh and S. Kishor, Some types of ${\eta}$-Ricci solitons on Lorentzian para-Sasakian manifolds, Facta Universitatis (NIS), 33(2) (2018), 217-230.
  6. A. M. Blaga, ${\eta}$-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30(2) (2016), 489-496. https://doi.org/10.2298/FIL1602489B
  7. B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York-London, 1983.
  8. D. G. Prakasha and B. S. Hadimani, ${\eta}$-Ricci solitons on para-Sasakian manifolds, J. Geom., 108 (2017), 383-392. https://doi.org/10.1007/s00022-016-0345-z
  9. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math/0211159, 2002, 1-39.
  10. G. Perelman, Ricci flow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, 2003, 1-22.
  11. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61(2) (2009), 205-212. https://doi.org/10.2748/tmj/1245849443
  12. K. Mandal and U. C. De, Quarter-symmetric metric connection in a P-Sasakian manifold, Analele Univ. de Vest, Timisoara Seria Matematica-Informatica, LIII(1) (2015), 137-150.
  13. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Math., Vol. 3, World Sci., 1984.
  14. M. Ahmad, J. B. Jun, and A. Haseeb, Hypersurfaces of almost r-paracontact Riemannian manifold with a quarter symmetric metric connection, Bull. Korean Math. Soc., 46(3) (2009), 477-487. https://doi.org/10.4134/BKMS.2009.46.3.477
  15. M. Ali and Z. Ahsan, Gravitational field of Schwarzschild soliton, Arab J. Math. Sci., 21 (2015), 15-21. https://doi.org/10.1016/j.ajmsc.2013.10.003
  16. M. M. Tripathi, E. Kilic, S. Y. Perktas and S. Keles, Indefinite almost para-contact metric manifolds, Int. J. Math. and Math. Sci., (2010), Article ID: 846195, 19 pages.
  17. R. Prasad and A. Haseeb, Conformal curvature tensor on K-contact manifolds with respect to the quarter symmetric metric connection, Facta Universitatis (NIS), Ser. Math. Inform., 32 (2017), 503-514.
  18. R. Prasad and V. Srivastava, On ${\epsilon}$-Lorentzian para-Sasakian manifolds, Commun. Korean Math. Soc., 27 (2012), 297-306. https://doi.org/10.4134/CKMS.2012.27.2.297
  19. R. Sharma, Certain results on K-contact and (K,${\mu}$)-contact manifolds, J. Geom., 89 (2008), 138-147. https://doi.org/10.1007/s00022-008-2004-5
  20. R. N. Singh and S. K. Pandey, On quarter-symmetric metric connection in an LP-Sasakian manifold, Thai J. Math., 12 (2014), 357-371.
  21. R. N. Singh, S. K. Pandey, G. Pandey and K. Tiwari, On a semi-symmetric metric connection in an ${\epsilon}$-Kenmotsu manifold, Commun. Korean Math. Soc., 29 (2014), 331-343. https://doi.org/10.4134/CKMS.2014.29.2.331
  22. R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity, Contemp. Math., 71, American Math. Soc., (1988), 237-262.
  23. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306. https://doi.org/10.4310/jdg/1214436922
  24. S. Deshmukh, H. Alodan and H. Al-Sodais, A note on Ricci solitons, Balkan J. Geom. Appl., 16 (2011), 48-55.
  25. S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor (N.S.), 29(1975), 249-254.
  26. U. C. De, Ricci soliton and gradient Ricci soliton on P-Sasakian manifolds, The Aligarh Bull. of Maths., 29 (2010), 29-34.
  27. U. C. De and A. Sarkar, On ${\epsilon}$-Kenmotsu manifold, Hardonic J., 32 (2009), 231-242.
  28. U. C. De and A. K. Mondal, Quarter-symmetric metric connection on 3-dimensional quasi-Sasakian manifolds, SUT J. Math., 46 (2010), 35-52.
  29. X. Xufeng and C. Xiaoli, Two theorems on ${\epsilon}$-Sasakian manifolds, Int. J. Math. Math. Sci., 21 (1998), 249-54. https://doi.org/10.1155/S0161171298000350