Some Semisymmetry Curvature Conditions on Paracontact Metric (k,\mu)-Manifolds

Abstract

The aim of the article is to study paracontact metric $(k,\mu)$-manifolds satisfying some semisymmetry curvature conditions. Also, we show that if a paracontact metric $(k,\mu)$-manifold is Ricci pseudo-symmetric then it is an Einstein manifold provided $k\neq 1$.                                                                                                                                                                                                                                                                                                                                                                         

Keywords

• Paracontact metric (k;)-manifolds
• h-projectively semisymmetry
• -projectively semisymmetry
• Ricci pseudo-symmetry

References

1. Cappelletti-Montano B., Erken Kupeli I., Murathan C., Nullity conditions in paracontact geometry, Dif. Geom. Appl., 30, 665-693, 2012.
2. Cappelletti-Montano B., Di Terlizzi L., Geometric structures associated to a contact metric (k,μ)-space, Pasific J. Math., 246(2), 257-292, 2010.
3. Cartan E., Sur une classes remarquable d'espaces de Riemann, Bull. Soc. Math. France, 54, 214-264, 1926.
4. Erken Kupeli I., Murathan C., A study of three-dimensional paracontact (k, μ, v)-spaces, Int. J. Geomet. Meth. Mod. Phys., 14(7), 2017.
5. Kaneyuki S., Williams F.L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99, 173-187, 1985.
6. Mandal K., De U.C., Paracontact metric (k,μ)-spaces satisfying certain curvature conditions, Kyungpook Math. J., 59(1) 163-174, 2019.
7. Shirokov P.A., Collected Works of Geometry, Kazan University Press, 1966.
8. Soos G., Uber die geodatischen Abbildungen von Riemannaschen Raumen auf projektiv symmetrische Riemannsche Raume, Acta. Math. Acad. Sci. Hungar. Tom., 9, 359-361, 1958.

9. Szab'o Z.I., Structure theorems on Riemannian spaces satisfying R(X,Y).R=0, the local version, J. Dif. Geometry, 17, 531-582, 1982.
10. Yano K., Bochner S., Curvature and Betti Numbers, Annals of Mathematics Studies, 32, Princeton University Press, 1953.
11. Yıldız A., De U.C., A classification of (k,μ)-contact metric manifolds, Commun. Korean Math. Soc., 27, 327-339, 2012.
12. Yıldız A., De U.C., Certain semisymmetry curvature conditions on paracontact metric (k,μ)-manifolds, Math. Sci. Appl. E-Notes, 8(1), 1-10, 2020.
13. Zamkovoy S., Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom., 36, 37-60, 2009.
14. Zamkovoy S., Tzanov V., Non-existence of at paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform., 100, 27-34, 2011.