Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Paracontact Metric (k, 𝜇)-spaces Satisfying Certain Curvature Conditions

  • Received : 2016.11.07
  • Accepted : 2018.06.08
  • Published : 2019.03.23
Download PDF
⟨ Previous Next ⟩

Abstract

The object of this paper is to classify paracontact metric ($k,{\mu}$)-spaces satisfying certain curvature conditions. We show that a paracontact metric ($k,{\mu}$)-space is Ricci semisymmetric if and only if the metric is Einstein, provided k < -1. Also we prove that a paracontact metric ($k,{\mu}$)-space is ${\phi}$-Ricci symmetric if and only if the metric is Einstein, provided $k{\neq}0$, -1. Moreover, we show that in a paracontact metric ($k,{\mu}$)-space with k < -1, a second order symmetric parallel tensor is a constant multiple of the associated metric tensor. Several consequences of these results are discussed.

Keywords

References

  1. D. V. Alekseevski, V. Cortes, A. S. Galaev and T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy, J. Reine Angew. Math., 635(2009), 23-69.
  2. D. V. Alekseevski, C. Medori and A. Tomassini, Maximally homogeneous para-CR manifolds, Ann. Global Anal. Geom., 30(2006), 1-27. https://doi.org/10.1007/s10455-005-9009-1
  3. D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91(1995), 189-214. https://doi.org/10.1007/BF02761646
  4. G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math., 55(2011), 697-718. https://doi.org/10.1215/ijm/1359762409
  5. B. Cappelletti-Montano, I. Kupeli Erken and C. Murathan, Nullity conditions in paracontact geometry, Differential Geom. Appl., 30(2012), 665-693. https://doi.org/10.1016/j.difgeo.2012.09.006
  6. V. Cortes, M. A. Lawn and L. Schafer, Affine hyperspheres associated to special para-Kahler manifolds, Int. J. Geom. Methods Mod. Phys., 3(2006), 995-1009. https://doi.org/10.1142/S0219887806001569
  7. U. C. De and A. Sarkar, On ${\phi}$-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc., 11(2008), 47-52.
  8. L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc., 25(1923), 297-306. https://doi.org/10.1090/S0002-9947-1923-1501245-6
  9. S. Erdem, On almost (para) contact (hyperbolic) metric manifolds and harmonicity of (${\phi}$, ${\phi}'$)-holomorphic maps between them, Huston J. Math., 28(2002), 21-45.
  10. S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99(1985), 173-187. https://doi.org/10.1017/S0027763000021565
  11. H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math., 27(1925), 91-98. https://doi.org/10.2307/1967964
  12. V. Martin-Molina, Local classification and examples of an important class of paracontact metric manifolds, Filomat, 29(2015), 507-515. https://doi.org/10.2298/FIL1503507M
  13. V. A. Mirzoyan, Structure theorems on Riemannian Ricci semisymmetic spaces (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 6(1992), 80-89.
  14. A. K. Mondal, U. C. De and C. Ozgur, Second order parallel tensors on (k, ${\mu}$)-contact metric manifolds, An. St. Univ. Ovidius Const. Ser. Mat., 18(2010), 229-238.
  15. D. G. Prakasha and K. K. Mirji, On ${\phi}$-symmetric N(k)-paracontact metric manifolds, J. Math., (2015), Art. ID 728298, 6 pp.
  16. R. Sharma, Second order parallel tensors on contact manifolds, Algebras Groups Geom., 7(1990), 145-152.
  17. R. Sharma, Second order parallel tensors on contact manifolds II, C.R. Math. Rep. Acad. Sci. Canada, 13(1991), 259-264.
  18. P. A. Shirokov, Constant vector fields and tensor fields of second order in Riemannian spaces, Izv. kazan Fiz. Mat. Obshchestva Ser., 25(1925), 86-114.
  19. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying $R(X,Y){\cdot}R=0$, the local version, J. Differential Geom., 17(1982), 531-582. https://doi.org/10.4310/jdg/1214437486
  20. T. Takahashi, Sasakian ${\phi}$-symmetic spaces, Tohoku Math. J., 29(1977), 91-113. https://doi.org/10.2748/tmj/1178240699
  21. S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom., 36(2009), 37-60. https://doi.org/10.1007/s10455-008-9147-3
  22. S. Zamkovoy and V. Tzanov, 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.