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Certain results on N(k)-quasi Einstein manifolds

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Abstract

The object of the present paper is to study the properties of N(k)-quasi Einstein manifolds. The existence of some classes of such manifolds are proved by constructing physical and geometrical examples. It is also shown that the characteristic vector field of the manifold is a unit parallel vector field as well as Killing vector field.

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References

  1. El Naschie, M.S.: Is Einstein general field equation more fundamental than quantum field theory and particle physics? Chaos Solitons Fractals 30(3), 759–764 (2006)

    Article  MathSciNet  Google Scholar 

  2. El Naschie, M.S.: Gödel universe, dualities and high energy particles in E-infinity. Chaos Solitons Fractals 25(3), 759–764 (2005)

    Article  MATH  Google Scholar 

  3. El Naschie, M.S.: A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos Solitons Fractals 19, 209–236 (2004)

    Article  MATH  Google Scholar 

  4. Chaki, M.C., Maithy, R.K.: On quasi Einstein manifolds. Publ. Math. Debr. 57(3–4), 297–306 (2000)

    MathSciNet  Google Scholar 

  5. Chaubey, S.K., Baisya, K.K., Siddiqi, M.D.: Existence of some classes of \(N(k)\)-quasi Einstein manifolds. Bol. Soc. Paran. Mat. (3s.) 38 (2020) (in editing)

  6. Debnath, P., Konar, A.: On quasi Einstein manifold and quasi Einstein space time. Differ. Geom. Dyn. Syst. 12, 73–82 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser Boston Inc, Boston (2002)

    Book  MATH  Google Scholar 

  8. Okumura, M.: Some remarks on space with a certain contact structure. Tohoku Math. J. 14, 135–145 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaki, M.C.: On generalized quasi Einstein manifolds. Publ. Math. Debr. 58(4), 683–691 (2001)

    MathSciNet  MATH  Google Scholar 

  10. De, U.C., Gazi, A.K.: On Pseudo Ricci Symmetric Manifolds. Analele Stiintifice Ale Universitataii “Al. I. CUZA” Iasi Matematica, Tomul LVIII, pp. 209–222 (2012)

  11. Guha, S.: On quasi Einstein and generalized quasi Einstein manifolds. Facta Univ. Ser. Mech. Autom. Control Robot 3(14), 821–842 (2003)

    MathSciNet  MATH  Google Scholar 

  12. De, U.C., Ghosh, G.C.: On quasi Einstein manifolds II. Bull. Calc. Math. Soc. 96(2), 135–138 (2004)

    MathSciNet  MATH  Google Scholar 

  13. De, U.C., Ghosh, G.C.: On quasi Einstein and special quasi Einstein manifolds. In: Proceedings of the International Conference of Mathematics and Its Applications. Kuwait University, April 5–7 (2004), pp. 178–191

  14. De, U.C., Ghosh, G.C.: On quasi Einstein manifolds. Period. Math. Hung. 48(1–2), 223–231 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Deszcz, R., Hotlos, M., Senturk, Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27, 375–389 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Mantica, C.A., Suh, Y.J.: Conformally symmetric manifolds and quasi conformally recurrent Riemannian manifolds. Balk. J. Geom. Appl. 16, 66–77 (2011)

    MathSciNet  MATH  Google Scholar 

  17. De, U.C., De, B.K.: On quasi Einstein manifolds. Commun. Korean Math. Soc. 23(3), 413–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tanno, S.: Ricci curvatures of contact Riemannian manifolds. Tohoku Math. J. 40, 441–417 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tripathi, M.M., Kim, J.S.: On \(N(k)\)-quasi Einstein manifolds. Commun. Korean Math. Soc. 22(3), 411–417 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Özgür, C̈., Tripathi, M.M.: On the concircular curvature tensor of an \(N(k)\)-quasi Einstein manifolds. Math. Pann. 18(1), 95–100 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Özgür, C̈., Sular, S.: On \(N(k)\)-quasi Einstein manifolds satisfying certain conditions. Balk. J. Geom. Appl. 13(2), 74–79 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Özgür, C̈.: \(N(k)\)-quasi Einstein manifolds satisfying certain conditions. Chaos Solitons Fractals 38(5), 1373–1377 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. De, A., De, U.C., Gazi, A.K.: On a class of \(N(k)\)-quasi Einstein manifolds. Commun. Korean Math. Soc. 26(4), 623–634 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mallick, S., De, U.C.: \(\cal{Z}\) tensor on \(N(k)\)-quasi Einstein manifolds. Kyungpook Math. J. 56, 979–991 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yildiz, A., De, U.C., Centinkaya, A.: \(N(k)\)-quasi Einstein manifolds satisfying certain curvature conditions. Bull. Malays. Math. Sci. Soc. 36(4), 1139–1149 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Singh, R.N., Pandey, M.K., Gautam, D.: On \(N(k)\)-quasi Einstein manifolds. Novi Sad J. Math. 40(2), 23–28 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Taleshian, A., Hosseinzadeh, A.A.: Investigation of some conditions on \(N(k)\)-quasi Einstein manifolds. Bull. Malays. Math. Sci. Soc. 34(3), 455–464 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Hosseinzadeh, A.A., Taleshian, A.: On conformal and quasi conformal curvature tensors of an \(N(k)\)-quasi Einstein manifolds. Commun. Korean Math. Soc. 27(2), 317–326 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, Y., Xu, S.: Some conditions of \(N(k)\)-quasi Einstein manifolds. Int. J. Dig. Cont. Tech. Appl. 6(8), 144–150 (2012)

    Google Scholar 

  30. Chaubey, S.K.: Existence of \(N(k)\)-quasi Einstein manifolds. Facta Univ. (NIS) Ser. Math. Inform. 32(3), 369–385 (2017)

    Article  MathSciNet  Google Scholar 

  31. Pokhariyal, G.P., Mishra, R.S.: Curvature tensor and their relativistic significance II. Yokohama Math. J. 19, 97–103 (1971)

    MathSciNet  MATH  Google Scholar 

  32. Ojha, R.H.: \(M\)-projectively flat Sasakian manifolds. Indian J. Pure Appl. Math. 17(4), 481–484 (1986)

    MathSciNet  MATH  Google Scholar 

  33. Ojha, R.H.: A note on the \(M\)-projective curvature tensor. Indian J. Pure Appl. Math. 8(12), 1531–1534 (1975)

    MathSciNet  MATH  Google Scholar 

  34. Chaubey, S.K., Ojha, R.H.: On the \(m\)-projective curvature tensor of a Kenmotsu manifold. Differ. Geom. Dyn. Syst. 12, 52–60 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Chaubey, S.K.: Some properties of \( LP\)-Sasakian manifolds equipped with \(m\)-projective curvature tensor. Bull. Math. Anal. Appl. 3(4), 50–58 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Chaubey, S.K.: On weakly \(m\)-projective symmetric manifolds. Novi Sad J. Math. 42(1), 67–79 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Prakasha, D.G., Zengin, F.O., Chavan, V.: On \({\cal{M}}\)-projectively semisymmetric Lorentzian \(\alpha \)-Sasakian manifolds. Afr. Mat. (2017). https://doi.org/10.1007/s13370-017-0493-9

    MathSciNet  MATH  Google Scholar 

  38. Mantica, C.A., Molinari, L.G.: Weakly Z-symmetric manifolds. Acta Math. Hung. 135, 80–96 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Harko, T., et al.: \(f(R, T)\)-gravity (2011). arXiv:1104.2669v2 [gr-qc]

  40. Capozziello, S., et al.: Quintessence without scalar fields. Recent Res. Dev. Astron. Astrophys. I 625 (2003)

  41. Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  42. Janssens, D., Vanheche, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)

    Article  MathSciNet  Google Scholar 

  44. Yano, K.: On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20, 340–345 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  45. Schouten, J.A.: Ricci-Calculus, An Introduction to Tensor Analysis and Geometrical Applications. Springer, Berlin (1954)

    MATH  Google Scholar 

  46. Chen, B.Y., Yano, K.: Hyper surfaces of a conformally flat space. Tensor N. S. 26, 318–322 (1972)

    MathSciNet  MATH  Google Scholar 

  47. Yano, K., Kon, M.: Structures on Manifolds. Series in Pure Mathematics. World Scientific Publishing Co, Singapore (1984)

    MATH  Google Scholar 

  48. Mantica, C.A., Suh, Y.J.: Recurrent Z forms on Riemannian and Kaehler manifolds. Int. J. Geom. Methods Mod. Phys. 9, 1250059 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wants to express his sincere thanks and gratitude to the Editor and anonymous referees for their valuable comments towards the improvement of the paper.

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  1. Section of Mathematics, Department of Information Technology, Shinas College of Technology, P.O. Box 77, 324, Shinas, Sultanate of Oman

    S. K. Chaubey

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Chaubey, S.K. Certain results on N(k)-quasi Einstein manifolds. Afr. Mat. 30, 113–127 (2019). https://doi.org/10.1007/s13370-018-0631-z

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