# ON Φ-RECURRENT (k, μ)-CONTACT METRIC MANIFOLDS

• Published : 2008.11.30
• 57 3

#### Abstract

In this paper we prove that a $\phi$-recurrent (k, $\mu$)-contact metric manifold is an $\eta$-Einstein manifold with constant coefficients. Next, we prove that a three-dimensional locally $\phi$-recurrent (k, $\mu$)-contact metric manifold is the space of constant curvature. The existence of $\phi$-recurrent (k, $\mu$)-manifold is proved by a non-trivial example.

#### Keywords

(k, $\mu$)-contact metric manifolds;$\eta$-Einstein manifold;$\phi$-recurrent (k, $\mu$)-contact metric manifolds

#### References

1. C. Baikoussis, D. E. Blair, and T. Koufogiorgos, A decomposition of the curvature tensor of a contact manifold satisfying $R(X,Y)\xi=\kappa(\eta(Y)X-\eta(X)Y)$, Mathematics Technical Report, University of Ioannina, 1992
2. D. E. Blair and H. Chen, A classification of 3-dimensional contact metric manifolds with $Q\phi = \phi{Q}$. II, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 4, 379-383
3. D. E. Blair, J. S. Kim, and M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42 (2005), no. 5, 883-892 https://doi.org/10.4134/JKMS.2005.42.5.883
4. D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), no. 1-3, 189-214 https://doi.org/10.1007/BF02761646
5. E. Boeckx, A full classification of contact metric ($\kappa,\,\mu$)-spaces, Illinois J. Math. 44 (2000), no. 1, 212-219
6. E. Boeckx, P. Buecken, and L. Vanhecke, $\phi$-symmetric contact metric spaces, Glasg. Math. J. 41 (1999), no. 3, 409-416 https://doi.org/10.1017/S0017089599000579
7. U. C. De and A. K. Gazi, On $\Phi$-recurrent N(k)-contact metric manifolds, Math. J. Okayama Univ. 50 (2008), 101-112
8. U. C. De, A. A. Shaikh, and S. Biswas, On $\Phi$-recurrent Sasakian manifolds, Novi Sad J. Math. 33 (2003), no. 2, 43-48
9. J.-B. Jun and U.-K. Kim, On 3-dimensional almost contact metric manifolds, Kyungpook Math. J. 34 (1994), no. 2, 293-301
10. B. J. Papantoniou, Contact Riemannian manifolds satisfying R($\xi$,X)R = 0 and $\xi\, \in$ (k, )-nullity distribution, Yokohama Math. J. 40 (1993), no. 2, 149-161
11. T. Takahashi, Sasakian ϕ-symmetric spaces, Tohoku Math. J. (2) 29 (1977), no. 1, 91-113 https://doi.org/10.2748/tmj/1178240699
12. D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. (2) 29 (1977), no. 3, 319-324 https://doi.org/10.2748/tmj/1178240602
13. S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J. (2) 40 (1988), no. 3, 441-448 https://doi.org/10.2748/tmj/1178227985
14. D. E. Blair, T. Koufogiorgos, and R. Sharma, A classification of 3-dimensional contact metric manifolds with $Q\phi = \phi{Q}$, Kodai Math. J. 13 (1990), no. 3, 391-401 https://doi.org/10.2996/kmj/1138039284

#### Cited by

1. On ϕ-quasiconformally symmetric (κ,μ)-contact manifolds vol.31, pp.4, 2010, https://doi.org/10.1134/S1995080210040086
2. CERTAIN SEMISYMMETRY PROPERTIES OF (𝜅, 𝜇)-CONTACT METRIC MANIFOLDS vol.53, pp.4, 2016, https://doi.org/10.4134/BKMS.b150638