Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON SOME CLASSES OF WEAKLY Z-SYMMETRIC MANIFOLDS

  • Received : 2019.08.06
  • Accepted : 2019.12.17
  • Published : 2020.07.31
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Abstract

The aim of the paper is to study some geometric properties of weakly Z-symmetric manifolds. Weakly Z-symmetric manifolds with Codazzi type and cyclic parallel Z tensor are studied. We consider Einstein weakly Z-symmetric manifolds and conformally flat weakly Z-symmetric manifolds. Next, it is shown that a totally umbilical hypersurface of a conformally flat weakly Z-symmetric manifolds is of quasi constant curvature. Also, decomposable weakly Z-symmetric manifolds are studied and some examples are constructed to support the existence of such manifolds.

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References

  1. A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-74311-8
  2. T. Q. Binh, On weakly symmetric Riemannian spaces, Publ. Math. Debrecen 42 (1993), no. 1-2, 103-107.
  3. E. Cartan, Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France 54 (1926), 214-264. https://doi.org/10.24033/bsmf.1105
  4. M. C. Chaki, On pseudo symmetric manifolds, An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Mat. 33 (1987), no. 1, 53-58.
  5. M. C. Chaki, On generalized quasi Einstein manifolds, Publ. Math. Debrecen 58 (2001), no. 4, 683-691.
  6. M. C. Chaki and B. Gupta, On conformally symmetric spaces, Indian J. Math. 5 (1963), 113-122.
  7. B. Chen and K. Yano, Hypersurfaces of a conformally flat space, Tensor (N.S.) 26 (1972), 318-322.
  8. S. Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126. https://doi.org/10.1007/BF02960745
  9. U. C. De, On weakly symmetric structures on a Riemannian manifold, Facta Univ. Ser. Mech. Automat. Control Robot. 3 (2003), no. 14, 805-819.
  10. U. C. De and S. Bandyopadhyay, On weakly symmetric Riemannian spaces, Publ. Math. Debrecen 54 (1999), no. 3-4, 377-381.
  11. U. C. De, C. A. Mantica, and Y. J. Suh, On weakly cyclic Z symmetric manifolds, Acta Math. Hungar. 146 (2015), no. 1, 153-167. https://doi.org/10.1007/s10474-014-0462-9
  12. U. C. De and P. Pal, On almost pseudo-Z-symmetric manifolds, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 53 (2014), no. 1, 25-43.
  13. U. C. De and J. Sengupta, On a weakly symmetric Riemannian manifold admitting a special type of semi-symmetric metric connection, Novi Sad J. Math. 29 (1999), no. 3, 89-95.
  14. U. C. De and A. A. Shaikh, Differential geometry of manifolds, Alpha Science International Ltd. Oxford, U.K., 263-272, 2007.
  15. A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978), no. 3, 259-280. https://doi.org/10.1007/BF00151525
  16. S. K. Hui, On weakly $W_3$-symmetric manifolds, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 50 (2011), no. 1, 53-71.
  17. 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
  18. 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
  19. F. Ozen and S. Altay, On weakly and pseudo-symmetric Riemannian spaces, Indian J. Pure Appl. Math. 33 (2002), no. 10, 1477-1488.
  20. M. Prvanovic, On weakly symmetric Riemannian manifolds, Publ. Math. Debrecen 46 (1995), no. 1-2, 19-25.
  21. J. A. Schouten, Ricci-calculus. An introduction to tensor analysis and its geometrical applications, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete, Bd X, Springer-Verlag, Berlin, 1954.
  22. L. Tamassy and T. Q. Binh, On weakly symmetric and weakly projective symmetric Riemannian manifolds, in Differential geometry and its applications (Eger, 1989), 663- 670, Colloq. Math. Soc. Janos Bolyai, 56, North-Holland, Amsterdam,1992.
  23. L. Tamassy and T. Q. Binh, On weak symmetries of Einstein and Sasakian manifolds, Tensor (N.S.) 53 (1993), Commemoration Volume I, 140-148.
  24. A. G. Walker, On Ruse's spaces of recurrent curvature, Proc. London Math. Soc. (2) 52 (1950), 36-64. https://doi.org/10.1112/plms/s2-52.1.36
  25. K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, 3, World Scientific Publishing Co., Singapore, 1984.