Lorentzian para-Sasakian Manifolds Admitting a New Type of Quarter-symmetric Non-metric ξ-connection

Year 2019, , 250 - 259, 03.10.2019

S. K. Chaubey , Uday Chand De

https://doi.org/10.36890/iejg.548364

Cited By: 6

Abstract

We define a new type of quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold and prove its existence. We provide its application in the general theory of relativity. To validate the existence of the quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold, we give a non-trivial example in dimension $4$ and verify our results.

Keywords

Lorentzian para-Sasakian manifold, quarter-symmetric non-metric connection, conformal curvature tensors, Einstein space

References

• Reference1K. Arslan, R. Deszcz, R. Ezentas, M. Hotlo$\acute{s}$ and C. Murathan, On generalized Robertson-Walker spacetimes satisfying some curvature condition, Turk J Math 38 (2014), 353-373.
• Reference2A. A. Aqeel, U. C. De and G. C. Ghosh, On Lorentzian para-Sasakian manifolds, Kuwait J. Sci. Eng. 31 (2) (2004), 1-13.
• Reference3A. Barman, Weakly symmetric and weakly Ricci-symmetric $LP$-Sasakian manifolds admitting a quarter-symmetric metric connection, Novi Sad J. Math. 45 (2) (2015), 143-153.
• Reference4E. Cartan, Sur une classe remarquable d’espaces de Riemannian, Bull. Soc. Math. France 54 (1926), 214-264.
• Reference5S. K. Chaubey and R. H. Ojha, On semi-symmetric non-metric and quarter-symmetric metric connections, Tensor N. S. 70 2 (2008), 202-213.
• Reference6U. C. De, K. Matsumoto and A. A. Shaikh, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Mat. de Messina(3) (1999), 149-156.
• Reference7U. C. De, L. Velimirovi$\acute{c}$ and S. Mallick, On a type of spacetime, International Journal of Geometric Methods in Modern Physics 14 (1) (2017) 1750003 (9 pages).
• Reference8S. A. Demirba$\breve{g}$, H. B. Yilmaz, S. A. Uysal and F. $\ddot{O}.$ Zengin, On quasi Einstein manifolds admitting a Ricci quarter-symmetric metric connection, Bull. of Math. Anal. and Appl., 3 (4), (2011), 84-91.
• Reference9G. F. R. Ellis, Relativistic Cosmology in ’General Relativity and Cosmology’, ed. R. K. Sachs (Academic Press, London, 1971).
• Reference10I. Eriksson and J. M. M. Senovilla, Note on (conformally) semi-symmetric spacetimes, Class. Quantum Grav. 27 027001 (2010).
• Reference11 A. Friedmann and J. A. Schouten, Uber die Geometry der halbsymmetrischen Ubertragung. Math Zeitschr 21 (1924), 211-223.
• Reference12S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor, N.S., 29 (1975), 249-254.
• Reference13S. G$\ddot{u}$ler and S. A. Demirba$\breve{g}$, A study of generalized quasi Einstein spacetimeswith applications in general relativity, Int J Theor Phys 55 (2016), 548–562.
• Reference14 H. A. Hayden, Subspace of space with torsion. Proc. London Math. Soc. 34, 27-50 (1932).
• Reference15C. A. Mantica, U. C. De, Y. J. Suh and L. G. Molinari, Perfect fluid spacetimes with harmonic generalized curvature tensor, Osaka J. Math.56 (2019), 173-182. Reference16K. Matsumoto, On Lorentzian para contact manifolds, Bull. Yamagata Univ. Nat. Sci. 12 (1989), 151-156.
• Reference17K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor N. S. 47 (1988), 189-197.
• Reference18I. Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, Classical Analysis, World Scientific Publi., Singapore (1992), 155-169.
• Reference19I. Mihai, U. C. De and A. A. Shaikh, On Lorentzian para-Sasakian manifolds, Korean J. Math. Sci. 6 (1999), 1-13.
• Reference20R. S. Mishra and S. N. Pandey, On quarter symmetric metric $F$-connections, Tensor N. S. 34 (1980), 1-7.
• Reference21C. Murathan, A. Yildiz, K. Arslan and U. C. De, On a class of Lorentzian para-Sasakian manifolds, Proc. Estonian Acad. Sci. Phys. Math. 55 (4) (2006), 210-219.
• Reference22 S. C. Rastogi, On quarter-symmetric metric connection, C. R. Acad. Bulg. Sci. 31 (8) (1978), 811-814.
• Reference23E. M. Patterson, Some theorems on Ricci-recurrent spaces, J. London Math. Soc. 27 (1952), 287-295.
• Reference24R. Prasad and A. Haseeb, On a Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection, Novi Sad J. Math.46 (2) (2016), 103-116.
• Reference25S. K. Srivastava, Scale factor dependent equation of state for curvature inspired dark energy,phantom barrier and late cosmic acceleration, Physics Letters B 643 (2006) 1-4.
• Reference26S. Sular, C. $\ddot{O}$zg$\ddot{u}$r and U. C. De, Quarter-symmetric metric connection in a Kenmotsu manifold, SUT Journal of Mathematics 44 (2008), 297-306.
• Reference27Z. I. Szabó, Structure theorems on Riemannian spaces satisfying $R(X,Y) \cdot R=0.$ $I$. The local version. J. Diff. Geom 17 (1982), 531-582.
• Reference28H. Weyl, Reine Infinitesimalgeometrie. Math. Z. 2 (1918), 384-411.
• Reference29F. $\ddot{O}$. Zengin, $M$-projectively flat spacetimes, Math. Rep. 4 (2012), no. 4, 363–370.