A Normal Paracontact Metric Manifold Satisfying Some Conditions on the $M$-Projective Curvature Tensor

Abstract

In the present paper we have studied the curvature tensors of a normal paracontact metric manifold satisfying the conditions $R(\xi,Y)W^{*}=0$, $W^{*}(\xi,Y)R=0$, $W^{*}(\xi,Y)\widetilde{Z}=0$, $W^{*}(\xi,Y)S=0$ and $W^{*}(\xi,Y)\widetilde{C}=0$, where $W^{*}$,$R$, $S$, $\widetilde{Z}$ and $\widetilde{C}$ are the $M$-projective curvature, Riemannian curvature, Ricci, concircular curvature and quasi-conformal curvature tensor, respectively.

Keywords

• Normal Paracontact Metric Manifold,M-Projective Curvature Tensor,Einstein Manifold

References

1. [1] Atçeken,M. and Yıldırım, U . (2016). AlmostC(a)􀀀Manifolds Satisfying Certain Curvature Conditions, Advanced Studies in ContemporaryMathematics, 26 (3), 567-578.
2. [2] Atçeken, M. and Yıldırım, U . (2016). On Almost C(a)􀀀Manifolds Satisfying Certain Conditions on Quasi-Conformal Curvature Tensor, Proceedings of the Jangjeon Mathematical Society, 19(1), 115-124.
3. [3] Chaubey, S.K. and Ojha, R.J. (2010). On the M-projecvite curvature tensor of a Kenmotsu manifold, Differential Geometry-Dynamical Systems, 12, 52-60.
4. [4] Chaubey, S.K. (2011). Some properties of LP-Sasakian manifolds equipped with M-projective curvature tensor, Bulletin of Mathematical Analysis and Applications, 3(4), 50-58.
5. [5] Chaubey, S.K., Prakash, S. and Nivas, R. (2012). Some properties of M-projective curvature tensor in Kenmotsu manifolds, Bulletin of Mathemaical Analysis and Applications, vol. 4 issue 3, 48-56.
6. [6] Devi, M.S. and Singh, J.P. (2015). On a type of M-projective curvature tensor in Kenmotsu manifolds, International J. of Math. Sci.and Eng. Appl., no.III, 37-49.
7. [7] Kenayuki, S. and Williams, F.L. (1985). Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. vol. 99, 173-187.
8. [8] Kumar, R. (2016). M-projective curvature tensor of a semi-symmetric metric connection in a Kenmotsu manifold, International Journal of Mathematics and its Applications, vol. 5, issue 1-A, 81-91.

9. [9] Ojha, R.H. (1986). M-projectively flat Sasakian manifolds, Indian J. Pure Appl. Math. 17(4), 481-484.
10. [10] Ojha, R.H. (1975). A note on the M-projective curvature tensor, Indian J. Pure Applied Math. 8(12), 1531-1534.
11. [11] Pokhariyal, G.P. and Mishra, R.S. (1971). Curvature tensor and their relativistic significance II, Yokohama Mathematical Journal, 19, 97-103.
12. [12] Singh, R.N. and Pandey, S.K. (2013). On the M-projective curvature tensor of N(k)-contact metric manifolds, ISRN Geometry, vol. 2013(2013).
13. [13] Singh, J.P. (2012). On M-projective recurrent Riemann manifold, Int. Journal of Math. Analysis, vol. 6, no. 24, 1173-1178.
14. [14] Vankatesha and Sumangala, B. (2013). On M-projective curvature tensor of a generalized Sasakian space form, Acta Math. Univ. Comenianae, vol. LXXXII, 2, 209-217.
15. [15] Welyczo, J. (2009). On Legendre curves in 3-dimensional normal almost paracontact metric manifolds, Result. Math. 54, 377-387.
16. [16] Welyczo, J. (2014). Slant curves in 3-dimensional normal almost paracontact metric manifolds, Mediterr. J. MAth. 11, 965-978.
17. [17] Yano, K. and Sawaki, S. (1968). Riemannian manifolds admitting a conformal transformation group. J. Differential Geom. 2, 161–184.
18. [18] Zamkovoy, S. (2009). Canonical connections on paracontact manifolds. Ann Glob. Anal. and Geom. 36, 37-60.