On the Geometry $(k,\mu )$-Paracontact Metric Manifold Satisfying Certain Curvature Conditions

Year 2021, Volume: 3 Issue: 1, 16 - 22, 30.08.2021

Pakize Uygun , Mehmet Atçeken

Abstract

In the present paper, we have studied the curvature tensors of (k,$\mu$)-paracontact metric manifold satisfying the conditions $\tilde{Z}(X,Y)\cdot R=0$, $\tilde{Z}$, \ $R(X,Y)\cdot \tilde{Z}=0$  and $R(X,Y)\cdot R=0$. According the cases, we have classified ($k,\mu$)-paracontact metric manifolds.

Keywords

($k,\mu$)-paracontact manifold, $\eta$-Einstein manifold, concircular curvature tensor, Riemannian curvature tensor

References

• Arslan, K., Murathan C. & Özgür C. (2000). On contact manifolds satisfying certain curvature conditions. An. Univ. Bucuresti Math., 49(2), 17-26.
• Atçeken, M. (2014). On generalized Sasakian space forms satisfying certain conditions on the concircular curvature tensor. Bull. Math. Anal. Appl., 6(1), 1-8.
• Atçeken, M. & Uygun P. (2020). Characterizations for totally geodesic submanifolds of (k;m)-paracontact metric manifolds. Korean J. Math., 28(3), 555-571.
• Calvaruso, G. (2011). Homogeneous paracontact metric three-manifolds, Illinois Journal of Mathematics, 55(2), 697-718.
• Cappelletti-Montano, B., Küpeli Erken, I. & Murathan C. (2012). Nullity conditions in paracontact geometry. Differential Geom. Appl., 30(6), 665-693.
• Kaneyuki, S., Williams, F. L. (1985). Almost paracontact and parahodge structures on manifolds. Nagoya Mathematical Journal, 99, 173-187.
• Kowalczyk, D. (2001). On some subclass of semisymmetric manifolds. Soochow J. Math., 27(4), 445-462.
• Mirzoyan, V. A. (1992). Structure theorems on Riemannian Ricci semisymmetric spaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 6, 80-89.
• Szabo, Z. I. (1982). Structure theorems on Riemannian spaces satisfying R(X;Y):R = 0;. I. The local version. Journal of Differential Geometry, 17(4), 531-582.
• Takahashi, T. (1977). Sasakian $\varphi$-symmetric spaces. Tohoku Math. J., 29(1), 91-113.
• Kon, M., & Yano, K. (1985). Structures on manifolds (Vol. 3). World scientific.
• Uygun P. & Atçeken M. (2021). On $(k,\mu )$-paracontact metric spaces satisfying some conditions on the $W_{0}^{\star }-$curvature tensor. NTMSCI, 9(2), 26-37.
• Yıldırım, Ü ., Atçeken, M. & Dirik, S. (2019). A normal paracontact metric manifold satisfying some conditions on the $M$-projective curvature tensor. Konuralp Journal of Mathematics, 7(1), 217-221.
• Yıldırım, Ü ., Atçeken, M. & Dirik, S. (2019). Pseudo projective curvature tensor satisfying some properties on a normal paracontact metric manifold. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1), 997-1006.
• Zamkovoy S. (2009). Canonical connections on paracontact manifolds. Ann. Global Anal. Geom., 36(1), 37-60.
• Zamkovoy S. & Tzanov V. (2011). Non-existence of flat paracontact metric structures in dimension greater than or equal to five. Annuaire Univ. Sofia Fac. Math. Inform., 100, 27-34.