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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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)
El Naschie, M.S.: Gödel universe, dualities and high energy particles in E-infinity. Chaos Solitons Fractals 25(3), 759–764 (2005)
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)
Chaki, M.C., Maithy, R.K.: On quasi Einstein manifolds. Publ. Math. Debr. 57(3–4), 297–306 (2000)
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)
Debnath, P., Konar, A.: On quasi Einstein manifold and quasi Einstein space time. Differ. Geom. Dyn. Syst. 12, 73–82 (2010)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser Boston Inc, Boston (2002)
Okumura, M.: Some remarks on space with a certain contact structure. Tohoku Math. J. 14, 135–145 (1962)
Chaki, M.C.: On generalized quasi Einstein manifolds. Publ. Math. Debr. 58(4), 683–691 (2001)
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)
Guha, S.: On quasi Einstein and generalized quasi Einstein manifolds. Facta Univ. Ser. Mech. Autom. Control Robot 3(14), 821–842 (2003)
De, U.C., Ghosh, G.C.: On quasi Einstein manifolds II. Bull. Calc. Math. Soc. 96(2), 135–138 (2004)
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
De, U.C., Ghosh, G.C.: On quasi Einstein manifolds. Period. Math. Hung. 48(1–2), 223–231 (2004)
Deszcz, R., Hotlos, M., Senturk, Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27, 375–389 (2001)
Mantica, C.A., Suh, Y.J.: Conformally symmetric manifolds and quasi conformally recurrent Riemannian manifolds. Balk. J. Geom. Appl. 16, 66–77 (2011)
De, U.C., De, B.K.: On quasi Einstein manifolds. Commun. Korean Math. Soc. 23(3), 413–420 (2008)
Tanno, S.: Ricci curvatures of contact Riemannian manifolds. Tohoku Math. J. 40, 441–417 (1988)
Tripathi, M.M., Kim, J.S.: On \(N(k)\)-quasi Einstein manifolds. Commun. Korean Math. Soc. 22(3), 411–417 (2007)
Özgür, C̈., Tripathi, M.M.: On the concircular curvature tensor of an \(N(k)\)-quasi Einstein manifolds. Math. Pann. 18(1), 95–100 (2007)
Özgür, C̈., Sular, S.: On \(N(k)\)-quasi Einstein manifolds satisfying certain conditions. Balk. J. Geom. Appl. 13(2), 74–79 (2008)
Özgür, C̈.: \(N(k)\)-quasi Einstein manifolds satisfying certain conditions. Chaos Solitons Fractals 38(5), 1373–1377 (2008)
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)
Mallick, S., De, U.C.: \(\cal{Z}\) tensor on \(N(k)\)-quasi Einstein manifolds. Kyungpook Math. J. 56, 979–991 (2016)
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)
Singh, R.N., Pandey, M.K., Gautam, D.: On \(N(k)\)-quasi Einstein manifolds. Novi Sad J. Math. 40(2), 23–28 (2010)
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)
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)
Yang, Y., Xu, S.: Some conditions of \(N(k)\)-quasi Einstein manifolds. Int. J. Dig. Cont. Tech. Appl. 6(8), 144–150 (2012)
Chaubey, S.K.: Existence of \(N(k)\)-quasi Einstein manifolds. Facta Univ. (NIS) Ser. Math. Inform. 32(3), 369–385 (2017)
Pokhariyal, G.P., Mishra, R.S.: Curvature tensor and their relativistic significance II. Yokohama Math. J. 19, 97–103 (1971)
Ojha, R.H.: \(M\)-projectively flat Sasakian manifolds. Indian J. Pure Appl. Math. 17(4), 481–484 (1986)
Ojha, R.H.: A note on the \(M\)-projective curvature tensor. Indian J. Pure Appl. Math. 8(12), 1531–1534 (1975)
Chaubey, S.K., Ojha, R.H.: On the \(m\)-projective curvature tensor of a Kenmotsu manifold. Differ. Geom. Dyn. Syst. 12, 52–60 (2010)
Chaubey, S.K.: Some properties of \( LP\)-Sasakian manifolds equipped with \(m\)-projective curvature tensor. Bull. Math. Anal. Appl. 3(4), 50–58 (2011)
Chaubey, S.K.: On weakly \(m\)-projective symmetric manifolds. Novi Sad J. Math. 42(1), 67–79 (2012)
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
Mantica, C.A., Molinari, L.G.: Weakly Z-symmetric manifolds. Acta Math. Hung. 135, 80–96 (2012)
Harko, T., et al.: \(f(R, T)\)-gravity (2011). arXiv:1104.2669v2 [gr-qc]
Capozziello, S., et al.: Quintessence without scalar fields. Recent Res. Dev. Astron. Astrophys. I 625 (2003)
Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)
Janssens, D., Vanheche, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)
Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)
Yano, K.: On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20, 340–345 (1944)
Schouten, J.A.: Ricci-Calculus, An Introduction to Tensor Analysis and Geometrical Applications. Springer, Berlin (1954)
Chen, B.Y., Yano, K.: Hyper surfaces of a conformally flat space. Tensor N. S. 26, 318–322 (1972)
Yano, K., Kon, M.: Structures on Manifolds. Series in Pure Mathematics. World Scientific Publishing Co, Singapore (1984)
Mantica, C.A., Suh, Y.J.: Recurrent Z forms on Riemannian and Kaehler manifolds. Int. J. Geom. Methods Mod. Phys. 9, 1250059 (2012)
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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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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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DOI: https://doi.org/10.1007/s13370-018-0631-z
Keywords
- Quasi Einstein
- k-nullity distribution
- N(k)-quasi Einstein
- Kenmotsu manifolds
- \(\mathcal {Z}\)-tensor
- \(f(r, T)\)-gravity
- Ricci soliton
- Torse forming vector field
- Curvature tensor