# NOTES ON WEAKLY CYCLIC Z-SYMMETRIC MANIFOLDS

• Kim, Jaeman (Department of Mathematics Education Kangwon National University)
• Received : 2016.11.21
• Accepted : 2017.06.08
• Published : 2018.01.31

#### Abstract

In this paper, we study some geometric structures of a weakly cyclic Z-symmetric manifold (briefly, $[W CZS]_n$). More precisely, we prove that a conformally flat $[W CZS]_n$ satisfying certain conditions is special conformally flat and hence the manifold can be isometrically immersed in an Euclidean manifold $E^n+1$ as a hypersurface if the manifold is simply connected. Also we show that there exists a $[W CZS]_4$ with one parameter family of its associated 1-forms.

#### References

1. T. Adati and T. Miyazawa, On a Riemannian space with recurrent conformal curvature, Tensor (N.S.) 18 (1967), 348-354.
2. J. K. Beem and P. E. Ehrilich, Global Lorentzian Geometry, Marcel Dekker Inc. New York, 1981.
3. T. Q. Binh and L. Tamassy, On weakly symmetric and weakly projectively symmetric Riemannian manifolds, Colloq. Math. Soc. Janos Bolyai 56 (1989), 663-670.
4. T. Q. Binh and L. Tamassy, On weakly symmetries of Einstein and Sasakian manifolds, Tensor (N.S.) 53 (1993), 140-148.
5. E. Cartan, Sur une classes remarquable d'espaces de Riemannian, Bull. Soc. Math. France 54 (1926), 214-264.
6. M. C. Chaki, On pseudo symmetric manifolds, An. cStiinct. Univ. Al. I. Cuza Iacsi Secct. I a Mat. 33 (1987), no. 1, 53-58.
7. M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulgr. J. Phys. 15 (1988), no. 6, 526-531.
8. M. C. Chaki and B. Gupta, On conformally symmetric spaces, Indian J. Math. 5 (1963), 113-122.
9. M. C. Chaki and S. Koley, On generalized pseudo Ricci symmetric manifolds, Period. Math. Hungar. 28 (1994), no. 2, 123-129. https://doi.org/10.1007/BF01876902
10. B. Y. Chen and K. Yano, Special conformally flat spaces and canal hypersurfaces, To-hoku Math. J. 25 (1973), 177-184. https://doi.org/10.2748/tmj/1178241376
11. U. C. De, C. A. Mantica, L. G. Molinari, and Y. J. Suh, On weakly cyclic Z-symmetric spacetimes, Acta Math. Hung. 149 (2016), no. 2, 462-477. https://doi.org/10.1007/s10474-016-0612-3
12. U. C. De, C. A. Mantica, and Y. J. Suh, On weakly cyclic Z-symmetric manifolds, Acta Math. Hungar. 146 (2015), 153-167. https://doi.org/10.1007/s10474-014-0462-9
13. S. K. Jana and A. A. Shaikh, On weakly cyclic Ricci symmetric manifolds, Ann. Polon. Math. 89 (2006), no. 3, 273-288. https://doi.org/10.4064/ap89-3-4
14. C. A. Mantica and L. G. Molinari, Weakly Z-symmetric manifolds, Acta Math. Hungar. 135 (2012), no. 1-2, 80-96. https://doi.org/10.1007/s10474-011-0166-3
15. C. A. Mantica and Y. J. Suh, Pseudo Z-symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 1, 1250004, 21 pp. https://doi.org/10.1142/S0219887812500041
16. C. A. Mantica and Y. J. Suh, Pseudo Z-symmetric space-times, J. Math. Phys. 55 (2014), no. 4, 042502, 12 pp. https://doi.org/10.1063/1.4871442
17. C. A. Mantica and Y. J. Suh, Pseudo Z-symmetric space-times with divergence-free Weyl tensor and ppwaves, Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 2, 1650015, 34 pp. https://doi.org/10.1142/S0219887816500158
18. A. G. Walker, On Ruse's space of recurrent curvature, Proc. London Math. Soc. 52 (1950), 36-54.
19. Y. Watanabe, Integral inequalities in compact orientable manifolds, Riemannian or Kahlerian, Kodai Math. Sem. Rep. 20 (1968), 264-271. https://doi.org/10.2996/kmj/1138845694