Ricci Solitons in (ε,δ)-Trans-Sasakian Manifolds

Main Article Content

C.S. Bagewadi, Gurupadavva Ingalahalli


We study Ricci solitons in (ε,δ)-trans-Sasakian manifolds. It is shown that a symmetric parallel second order covariant tensor in a (ε,δ)-trans-Sasakian manifold is a constant multiple of the metric tensor. Using this it is shown that if Vg + 2S is parallel where V is a given vector field, then (g,V) is Ricci soliton. Further, by virtue of this result, Ricci solitons for 3-dimensional (ε,δ)-trans-Sasakian manifolds are obtained. Also an example of Ricci solitons in 3-dimensional (ε,δ)-trans-Sasakian manifold is provided in the region where trans-Sasakian manifold is expanding (shrinking) the Lorentzian trans-Sasakian manifold is shrinking (expanding).

Article Details


  1. Amalendu Ghosh and R. Sharma, K-contact metrics as Ricci solitons, Beitr. Algebra Geom. 53 (1) (2012), 25-30.
  2. C.S. Bagewadi and Venkatesha, Some Curvature Tensors on a Trans-Sasakian Manifold, Turk. J. Math. 31 (2007), 111-121.
  3. C.S. Bagewadi and Gurupadavva Ingalahalli, Ricci Solitons in Lorentzian α-Sasakian Manifolds, Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 28(1) (2012), 59-68.
  4. A. Bejancu and K.L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, Int. J. Math and Math Sci., 16(3) (1993), 545-556.
  5. D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, 509, Springer-Verlag, berlin-New-York, (1976).
  6. D.E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat. 34 (1990), 199-207.
  7. Constantin Calin and Mircea Crasmareanu, From the Eisenhart Problem to Ricci Solitons in f-Kenmotsu Manifolds, Bull. Malays. Math. Sci. Soc. 33(3) (2010), 361-368.
  8. U.C. De and Avijit Sarkar, On ε-Kenmotsu manifolds, Hadronic J. 32 (2009), 231-242.
  9. U.C. De and M.M. Tripathi, Ricci tensor in 3-dimensional Trans-Sasakian manifolds, Kyungpook Math. J., 43(2) (2003), 247-255.
  10. K.L. Duggal, Space time manifolds and contact structures, Int. J. Math and Math Sci., 13(3) (1990), 545-553.
  11. L.P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc., 25(2) (1923), 297-306.
  12. A. Gray and L.M. Harvella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123(4) (1980), 35-58.
  13. Gurupadavva Ingalahalli and C.S. Bagewadi, Ricci solitons in (ε)-Trans-Sasakain manifolds, J. Tensor Soc. 6 (1) (2012), 145-159.
  14. R.S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity, (Santa Cruz. CA, 1986), Contemp. Math. 71, Amer. Math. Soc., (1988), 237-262.
  15. R.S. Hamilton, Three manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982), 255-306.
  16. T. Ivey, Ricci solitons on compact three-manifolds, Differ. Geom. Appl. 3 (1993), 301-307.
  17. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24(2) (1972), 93-103.
  18. R. Kumar, R. Rani and R.K. Nagaich, On sectional curvature of (ε)-Sasakian manifolds, Int. J. Math. Math. Sci. 2007 (2007) Article ID 93562, doi:10.1155/2007/93562.
  19. H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math. 27(2) (1925), 91-98.
  20. J.C. Marrero, The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl. 162 (1992), 77-86.
  21. Mircea Crasmareanu, Parallel tensors and Ricci Solitons in N(k)-Quasi Einstein Manifolds, Indian J. Pure Appl. Math., 43(4) (2012), 359-369.
  22. H.G. Nagaraja, C.R. Premalatha and G. Somashekhara, On (ε,δ)-Trans-Sasakian Strucutre, Proc. Est. Acad. Sci. 61 (1), (2012), 20-28.
  23. H.G. Nagaraja and C.R. Premalatha, Ricci solitons in Kenmotsu manifolds, J. Math. Anal. 3 (2), (2012), 18-24.
  24. J.A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), 187-193.
  25. A.A. Shaikh, K. K. Baishya and Eyasmin, On D-homothetic deformation of trans-Sasakian structure, Demonstr. Math., 41 (1) (2008), 171-188.
  26. R. Sharma, Second order parallel tensor in real and complex space forms, Internat. J. Math. Math. Sci., 12(4) (1989), 787-790.
  27. R. Sharma, Certain results on K-contact and (k, µ)-contact manifolds, J. Geom., 89(1-2) (2008), 138-147.
  28. S.S. Shukla and D.D. Singh, On (ε)-Trans-Sasakian manifolds, Int. J. Math. Anal. 49(4) (2010), 2401-2414.
  29. M.M. Tripathi, Ricci solitons in contact metric manifolds, arXiv:0801.4222 [math.DG].
  30. M. Turan, U.C. De and A. Yildiz, Ricci solitons and gradient Ricci solitons on 3-dimensional trans-Sasakian manifolds, Filomat, 26(2) (2012), 363-370.
  31. X. Xufeng and C. Xiaoli, Two theorems on (ε)-Sasakain manifolds, Int. J. Math. Math.Sci., 21(2) (1998), 249-254.