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ON 3-DIMENSIONAL LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS

  • Received : 2018.02.02
  • Accepted : 2018.05.03
  • Published : 2019.01.31

Abstract

The aim of the present paper is to study the Eisenhart problems of finding the properties of second order parallel tensors (symmetric and skew-symmetric) on a 3-dimensional LCS-manifold. We also investigate the properties of Ricci solitons, Ricci semisymmetric, locally ${\phi}$-symmetric, ${\eta}$-parallel Ricci tensor and a non-null concircular vector field on $(LCS)_3$-manifolds.

Keywords

References

  1. C. L. Bejan and M. Crasmareanu, Ricci solitons in manifolds with quasi-constant curvature, Publ. Math. Debrecen 78 (2011), no. 1, 235-243. https://doi.org/10.5486/PMD.2011.4797
  2. C. L. Bejan and M. Crasmareanu, Parallel second-order tensors on Vaisman manifolds, Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 2, 1750023, 8 pp.
  3. S. Brendle and R. Schoen, Curvature, sphere theorems, and the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 1, 1-32. https://doi.org/10.1090/S0273-0979-2010-01312-4
  4. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 361-368.
  5. M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulgar. J. Phys. 15 (1988), no. 6, 526-531.
  6. S. Chandra, S. K. Hui, and A. A. Shaikh, Second order parallel tensors and Ricci solitons on $(LCS)_n$-manifolds, Commun. Korean Math. Soc. 30 (2015), no. 2, 123-130. https://doi.org/10.4134/CKMS.2015.30.2.123
  7. M. Crasmareanu, Parallel tensors and Ricci solitons in N(k)-quasi Einstein manifolds, Indian J. Pure Appl. Math. 43 (2012), no. 4, 359-369. https://doi.org/10.1007/s13226-012-0022-3
  8. L. Das, Second order parallel tensors on ${\alpha}$-Sasakian manifold, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 23 (2007), no. 1, 65-69.
  9. L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc. 25 (1923), no. 2, 297-306. https://doi.org/10.1090/S0002-9947-1923-1501245-6
  10. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. https://doi.org/10.4310/jdg/1214436922
  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.
  12. S. K. Hui, On $\phi$-pseudo symmetries of $(LCS)_n$-manifolds, Kyungpook Math. J. 53 (2013), no. 2, 285-294. https://doi.org/10.5666/KMJ.2013.53.2.285
  13. S. K. Hui and M. Atceken, Contact warped product semi-slant submanifolds of $(LCS)_n$-manifolds, Acta Univ. Sapientiae Math. 3 (2011), no. 2, 212-224.
  14. S. K. Hui, M. Atceken, and S. Nandy, Contact CR-warped product submanifolds of $(LCS)_n$-manifolds, Acta Math. Univ. Comenian. (N.S.) 86 (2017), no. 1, 101-109.
  15. S. K. Hui, M. Atceken, and T. Pal, Warped product pseudo slant submanifolds $(LCS)_n$-manifolds, New Trends in Math. Sci. 5 (2017), 204-212.
  16. S. K. Hui and D. Chakraborty, $\eta$-Ricci solitons on $\eta$-Einstein $(LCS)_n$-manifolds, Acta Univ. Palacki. Olomuc., Fac. Rer. Nat. Mathematica 55 (2016), no. 2, 101-109.
  17. S. K. Hui and D. Chakraborty, Some types of Ricci solitons on $(LCS)_n$-manifolds, Journal of Mathematical Sciences: Advances and Applications 37 (2016), 1-17.
  18. S. K. Hui, R. S. Lemence, and D. Chakraborty, Ricci solitons on Ricci pseudosymmetric $(LCS)_n$-manifolds, arXiv:1707.03618v1 [math.DG] 12 Jul 2017.
  19. S. K. Hui, R. Prasad, and T. Pal, Ricci solitons on submanifolds of $(LCS)_n$-manifolds, to appear in Ganita, Bharat Ganit Parishad.
  20. X. Jia, Second order parallel tensors on quasi-constant curvature manifolds, Chinese Quart. J. Math. 17 (2002), no. 2, 101-105. https://doi.org/10.3969/j.issn.1002-0462.2002.02.019
  21. M. Kon, Invariant submanifolds in Sasakian manifolds, Math. Ann. 219 (1976), no. 3, 277-290. https://doi.org/10.1007/BF01354288
  22. H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. of Math. (2) 27 (1925), no. 2, 91-98. https://doi.org/10.2307/1967964
  23. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Natur. Sci. 12 (1989), no. 2, 151-156.
  24. K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor (N.S.) 47 (1988), no. 2, 189-197.
  25. B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
  26. G. Perelman, The entropy formula for the Ricci ow and its geometric applications, arXiv:math/0211159 [Math.DG] 2002 (2002), 1-39.
  27. G. Perelman, Ricci flow with surgery on three manifolds, arXiv:math/0303109 [Math.DG] 2003 (2013), 1-22.
  28. A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Math. J. 43 (2003), no. 2, 305-314.
  29. A. A. Shaikh, Some results on $(LCS)_n$-manifolds, J. Korean Math. Soc. 46 (2009), no. 3, 449-461. https://doi.org/10.4134/JKMS.2009.46.3.449
  30. A. A. Shaikh and H. Ahmad, Some transformations on $(LCS)_n$-manifolds, Tsukuba J. Math. 38 (2014), no. 1, 1-24. https://doi.org/10.21099/tkbjm/1407938669
  31. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat. 1 (2005), no. 2, 129-132. https://doi.org/10.3844/jmssp.2005.129.132
  32. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes - II, Amer. J. App. Sci. 3 (2006), no. 4, 1790-1794. https://doi.org/10.3844/ajassp.2006.1790.1794
  33. A. A. Shaikh, T. Basu, and S. Eyasmin, On locally $\phi$-symmetric $(LCS)_n$-manifolds, Int. J. Pure Appl. Math. 41 (2007), no. 8, 1161-1170.
  34. A. A. Shaikh, T. Basu, and S. Eyasmin, On the existence of $\phi$-recurrent $(LCS)_n$-manifolds, Extracta Math. 23 (2008), no. 1, 71-83.
  35. A. A. Shaikh and T. Q. Binh, On weakly symmetric $(LCS)_n$-manifolds, J. Adv. Math. Stud. 2 (2009), no. 2, 103-118.
  36. A. A. Shaikh and U. C. De, On 3-dimensional LP-Sasakian manifolds, Soochow J. Math. 26 (2000), no. 4, 359-368.
  37. A. A. Shaikh and S. K. Hui, On generalized $\phi$-recurrent $(LCS)_n$-manifolds, AIP Conference Proceedings 1309 (2010), 419-429.
  38. A. A. Shaikh, Y. Matsuyama, and S. K. Hui, On invariant submanifolds of $(LCS)_n$-manifolds, J. Egyptian Math. Soc. 24 (2016), no. 2, 263-269. https://doi.org/10.1016/j.joems.2015.05.008
  39. R. Sharma, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci. 12 (1989), no. 4, 787-790. https://doi.org/10.1155/S0161171289000967
  40. R. Sharma, Second order parallel tensors on contact manifolds, Algebras Groups Geom. 7(1990), no. 2, 145-152.
  41. R. Sharma, Second order parallel tensors on contact manifolds. II, C. R. Math. Rep. Acad.Sci. Canada 13 (1991), no. 6, 259-264.
  42. R. Sharma, On the curvature of contact metric manifolds, J. Geom. 53 (1995), no. 1-2,179-190. https://doi.org/10.1007/BF01224050
  43. H. Singh and Q. Khan, On special weakly symmetric Riemannian manifolds, Publ.Math. Debrecen 58 (2001), no. 3, 523-536.
  44. T. Takahashi, Sasakian ${\phi}$-symmetric spaces, Tohoku Math. J. (2) 29 (1977), no. 1,91-113. https://doi.org/10.2748/tmj/1178240699
  45. Y. Wang and X. Liu, Second order parallel tensors on almost Kenmotsu manifoldssatisfying the nullity distributions, Filomat 28 (2014), no. 4, 839-847. https://doi.org/10.2298/FIL1404839W
  46. S. K. Yadav, S. K. Chaubey, and D. L. Suthar, Some geometric properties of -Riccisolitons and gradient Ricci solitons on $(LCS)_n$-manifolds, CUBO: A Math. J. 19 (2017),no. 2, 32-48.
  47. S. K. Yadav, P. K. Dwivedi, and D. Suthar, On $(LCS)_{2n+1}$-manifolds satisfying certainconditions on the concircular curvature tensor, Thai J. Math. 9 (2011), no. 3, 597-603.
  48. S. K. Yadav, D. L. Suthar, and M. Hailu, On extended generalized ${\phi}$-recurrent $(LCS)_{2n+1}$-manifolds, Bol. Soc. Pran. Mat. (3s.) 37 (2019), no. 2, 9-21.
  49. K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo16 (1940), 195-200. https://doi.org/10.3792/pia/1195579139