A Study on Lorentzian $\alpha -$Sasakian Manifolds

Year 2019, Volume: 7 Issue: 2, 324 - 332, 15.10.2019

Rajendra Prasad Shashikant Pandey Sandeep Kumar Verma Sumeet Kumar

Abstract

The object of the present paper is to study the geometric properties of Concircular curvature tensor on Lorentzian $\alpha -$Sasakian manifold admitting a type of quarter-symmetric metric connection. In the last, we provide an example of 3-dimensional Lorentzian $\alpha -$Sasakian manifold endowed with the quarter-symmetric metric connection which is under consideration is an $\eta -$Einstein manifold with respect to the quarter-symmetric metric connection.

Keywords

Lorentzian $\alpha $-Sasakian manifolds, Quarter-symmetric metric connection, Concircular curvature tensor, $\eta -$Einstein manifold

References

• [1] A. Barman and G. Ghosh, Concircular curvature tensor of a semi-symmetric non-metric connection on P-Sasakian manifolds, An. Univ. Vest. Timis. Ser. Mat. Inform. LIV(2016), 47–58.
• [2] D. E. Blair, Inversion theory and conformal mapping, Stud. Math. Libr. 9, Amer. Math. Soc. (2000).
• [3] U. C. De and Krishnendu De, On Lorentzian Trans-Sasakian manifolds, Cummun. Fac. Sci. Univ. Ank. Series 62 (2013) no. 2, 37􀀀51:
• [4] A. Friedmann and J. A. Schouten, ¨U ber die Geometric der halbsymmetrischen ¨U bertragung, Math. Z.21(1924), 211–223.
• [5] S. Golab, On semi􀀀symmetric and quarter􀀀symmetric linear connections, Tensor (N.S.) 29 (1975) ; 249􀀀254.
• [6] K. Matsumoto, On Lorentzian para-contact manifold, Bull Yomagata Univ. Natur. Sci. 12 no. 2 (1989) 151-156.
• [7] W. Kuhnel, Conformal Transformations between Einstein Spaces, Bonn, 1985/1986, 105–146, Aspects Math. E12, Vieweg, Braunschweig, 1988.
• [8] A.K. Mondal, U.C. De, Some properties of a quarter-symmetric metric connection on a Sasakian manifold. Bull. Math. Analysis Appl. 3 (2009), 99-108.
• [9] K. Mandal and U.C. De, Quarter-symmetric metric connection in a P-Sasakian manifold. An. Univ. Vest. Timis. Ser. Math-Inform. LIII(2015), 137-150.
• [10] O’Neill, B., Semi-Riemannian geometry with applications to relativity, Academic press (1983).
• [11] C. Ozgur, f-conformally flat Lorentzian para-Sasakian manifolds. Radovi Matematicki 12 (2003), 99-106.
• [12] R. Prasad and A. Haseeb, On a Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection, Novi Sad J. Math. Vol. 46, No. 2, 2016, 103-116.
• [13] K. Yano, Concircular Geometry I. Concircular Transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195–200.
• [14] K. Yano and S. Bochner, Curvature and Betti numbers, Ann. of Math. Stud. 32(1953).
• [15] A. Yildiz and C. Murathan, On Lorentzian a-Sasakian manifolds. Kyungpook Math. J. 45 (2005),95-103:
• [16] A. Yildiz, M. Turan and B. E. Acet, On three dimensional Lorentzian a-Sasakian manifolds. Bull. Math. Anal. Appl. 1(2009), 90-98.
• [17] A. Yildiz, M. Turan and C. Murathan, A class of Lorentzian a-Sasakian manifolds. Kyungpook Math. J. 49 (2009),789-799.