On $(k,\mu )$-Paracontact Manifold Satisfying Some Curvature Conditions

Year 2023, Volume: 15 Issue: 1, 171 - 179, 30.06.2023

Pakize Uygun , Mehmet Atçeken

https://doi.org/10.47000/tjmcs.1153650

Cited By: 1

Abstract

In this work, we studied the curvature tensors of (k,$\mu$) satisfying the conditions $\widetilde{Z}(\xi ,\alpha _{3})\cdot P=0$, $\widetilde{Z}(\xi ,\alpha _{3})\cdot S=0$, $R(\xi ,\alpha _{3})\cdot P=0$, $R(\xi ,\alpha _{3})\cdot S=0$ and $P\cdot C=0$. Besides this, we classify $(k,\mu)$-paracontact manifolds. Also we researched conformally flat and $\phi $-conformally flat a $(k,\mu )-$paracontact metric manifolds.

Keywords

$(k,\mu)$-Paracontact manifold , Einstein manifold , Weyl projective curvature tensor , conformal curvature tensor , concircular curvature tensor

References

• Arslan, K., Murathan, C., Özgür, C., On contact manifolds satisfying certain curvature conditions, an. Univ. Bucuresti Math. 49(2)(2000), 17-26.
• Atçeken, M., On generalized Sasakian space forms satisfying certain conditions on the concircular curvature tensor, Bulletin of Mathematical Analysis and Applications, 6(1)(2014), 1-8.
• Atçeken, M., Uygun, P., Characterizations for totally geodesic submanifolds of $(k,\mu )$-paracontact metric manifolds, Korean J. Math. 28(2020), 555-571.
• Ayar, G., Chaubey, S.K., M-projective curvature tensor over cosymplectic manifolds, Differential Geometry - Dynamical Systems, 21(2019), 23-33.
• Calvaruso, G., Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55(2011), 697-718.
• Cappelletti-Montano, B., Küpeli Erken, I., Murathan, C., Nullity conditions in paracontact geometry, Differential Geom. Appl., 30(2012), 665-693.
• De, K., De, U.C., Conharmonic curvature tensor on Kenmotsu manifolds, Bulletin of the Transilvanya Univ., of Brasov, 6(55)(1)(2013).
• De, U.C., Sarkar, A., On the Projective curvature tensor of Generalized Sasakians-space forms, Questiones Mathematicae, 33(2010), 245-252.
• Kaneyuki, S., Williams, F.L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99(1985), 173-187.
• Kowalezyk, D., On some subclass of semisymmetric manifolds, Soochow J. Math., 27(2001), 445-461.
• Mert, T., Characterization of some special curvature tensor on Almost $C(a)$-manifold, Asian Journal of Math. and Computer Research, 29(1)(2022), 27-41.
• Mert, T., Atçeken, M., Almost $C(a)$-manifold on $W_{0}^{\star }-$curvature tensor, Applied Mathematical Sciences, 15(15)(2021), 693-703.
• Mert, T., Atçeken, M., Some important properties of almost Kenmotsu $\ (k,\mu ,\nu )-$space on the concircular curvature tensor, Fundamental Journal of Mathematics and Applications, 6(1)(2023), 51-60.
• Mirzoyan, V.A., Structure theorems on Riemannian Ricci semisymmetric spaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 6(1992), 80-89.
• Murathan, C., Erken, I.K., The harmonicity of the Reeb vector dield on paracontact metric 3-manifolds, arXiv:1305.1511v3 [math.DG], 27(2013).
• Özgür, C., De, U.C., On the quasi-conformal curvature tensor of a Kenmotsu manifold, Mathematica Pannonica, 17(2)(2006), 221-228.
• Szabo, Z.I., Structure theorems on Riemannian sp-aces satisfing $R(\alpha _{4},\alpha _{5}).R=0,$ I: The local version, J. Differential Geom., 17(4)(1982), 531-582.
• Takahashi, T., Sasakian $\phi -$symmetric spaces, Tohoku Math. J., 29(1977), 91-113.
• Yano, K., Kon, M., Structures Manifolds, Singapore, World Scientific, 1984.
• Yano, K., Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom. 2(1968), 161-184.
• Yıldız, A., De, U.C., Murathan, C., Arslan, K., On the weyl projective curvature tensor of an N(k)-contact metric manifold, Mathematica Pannonica, 21(1)(2010),1-14.
• Zamkovoy, S., Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36(2009), 37-60.
• Zamkovoy, S., Tzanov, V., Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform., 100(2011), 27-34.
• Zhen, G., On conformal symmetric K-contact manifolds, Chinese Quart. J. of Math., 7(1992), 5-10.