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On weakly cyclic Z symmetric manifolds

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Abstract

The object of the present paper is to study weakly cyclic Z symmetric manifolds. Some geometric properties have been studied. We obtain a sufficient condition for a weakly cyclic Z symmetric manifold to be a quasi Einstein manifold. Next we consider conformally flat weakly cyclic Z symmetric manifolds. Then we study Einstein (WCZS) n (n > 2). Also we study decomposable (WCZS) n (n > 2). We prove the equivalency of semisymmetry and Weyl-semisymmetry in a (WCZS) n (n > 2). Finally, we give an example of a (WCZS)4.

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References

  1. T. Adati and T. Miyazawa, On a Riemannian space with recurrent conformal curvature, Tensor (N.S.), 18 (1967), 348–354.

  2. C. L. Bejan and M. Crasmareanu, Weakly-symmetry of the Sasakian lifts on the tangent bundles, Publ. Math. Debrecen, 83 (2013), 63–69.

  3. T. Q. Binh, On weakly symmetric Riemannian spaces, Publ. Math. Debrecen, 42 (1993), 103–107.

  4. E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of Conullity Two, Singapore World Scientific Publishing, River Edge (1996).

  5. Cartan E.: Sur une classes remarquable d’espaces de Riemannian, Bull. Soc. Math. France, 54, 214–264 (1926)

    MathSciNet  Google Scholar 

  6. M. C. Chaki and B. Gupta, On conformally symmetric spaces, Indian J. Math., 5 (1963), 113–295.

  7. M. C. Chaki, On pseudo symmetric manifolds, Ann. St. Univ. “Al I Cuza” Iasi, 33 (1987), 53–58.

  8. M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulg. J. Phys., 15 (1988), 525– 531.

  9. M. C. Chaki and S. K. Saha, On pseudo-projective Ricci symmetric manifolds, Bulg. J. Phys. 21 (1994), 1–7.

  10. B. Y. Chen and K. Yano, Hypersurfaces of a conformally flat spaces, Tensor, N. S., 26 (1972), 318–322.

  11. Chern S.S.: On the curvature and characteristic classes of a Riemannian manifold, Abh. Math. Sem. Univ. Hamburg, 20, 117–126 (1956)

    Article  MathSciNet  Google Scholar 

  12. R. Couty, Sur les transformation defines par le groupe d’holonomie infinitesimale, C.R. Acad. Sci., 244 (1957), 553–555.

  13. U. C. De, On weakly symmetric structures on Riemannian manifolds, Facta Univ. Ser. Mech. Automat. Control Robot., 3 (2003), 805–819.

  14. U. C. De and S. Bandyopadhyay, On weakly symmetric spaces, Publ. Math. Debrecen, 54 (1999), 377–381.

  15. U. C. De and S. Bandyopadhyay, On weakly symmetric spaces, Acta Math. Hungar., 83 (2000), 205–212.

  16. 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), 89–95.

  17. L. P. Eisenhart, Riemannian Geometry, Princeton University Press (1949).

  18. A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259–280.

  19. S. K. Hui, On weakly W3-symmetric manifolds, Acta Univ. Palacki. Olomuc. Fac. rer. nat., Mathematica, 50 (2011), 53–71.

  20. O. Kowalski, An explicit classification of 3-dimensional Riemannian spaces satisfying R(X, Y).R = 0. Czechoslov. Math. J., 46 (1996), 427–474.

  21. A. Lichnerowich, Courbure numberes de Betti el Riemannian, Bull. Soc. Math. France, 54 (1950), 214–264.

  22. C. A. Mantica and L. G. Molinari, Weakly Z symmetric manifolds, Acta Math. Hungar., 135 (2012), 80–96.

  23. C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Meth. Mod. Phys., 9 (2012) 1250004 (21 pages).

  24. C. A. Mantica and Y. J. Suh, Recurrent Z forms on Riemannian and Kaehler manifolds, Int. J. Geom. Meth. Mod. Phys., 9 (2012) 1250059 (26 pages).

  25. C. A. Mantica and Y. J. Suh, Pseudo-Z symmetric space-times, J. Math Phys, 55, 042502 (2014), 12 pages

  26. B. O’Neill, Semi-Riemannian Geometry with Applications to the Relativity, Academic Press (New York–London, 1983).

  27. F. Ozen and S. Altay, Weakly and pseudo-symmetric Riemannian spaces, Indian J. Pure Appl. Math., 33 (2002), 1477–1488.

  28. F. Ozen and S. Altay, On weakly and pseudo-concircular symmetric structures on a Riemannian manifold, Acta Univ. Palack. Olomuc., Fac. Rerum. natur. Math., 47 (2008), 129–138.

  29. M. Prvanović, On weakly symmetric Riemannian manifolds, Publ. Math. Debrecen, 46 (1995), 19–25.

  30. J. A. Schouten, Ricci-Calculus, An Introduction to Tensor Analysis and its Geometrical Applications, Springer-Verlag (Berlin–G¨ottingen–Heidelberg, 1954).

  31. N. S. Sinyukov, Semisymmetric Riemannian spaces, in: Pervaya Vsesoyuznaya Geom. Konfer., Tezisy Dokl Kiev, 84 (1962) (in Russian).

  32. Z. I. Szabó, Classification and construction of complete hypersurfaces satisfying R(X, Y).R = 0, Acta Sci. Math. (Szeged), 47 (1981), 321–348.

  33. Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y).R = 0, I. The local version, J. Diff. Geom., 17 (1982), 531–582.

  34. Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y).R = 0, II. Global version, Geom. Dedicata, 19 (1985), 65–108.

  35. L. Tamássy and T. Q. Binh, On weakly symmetric and weakly projectively symmetric Riemannian manifolds, Colloq. Math. Soc. János Bolyai, 56 (1989), 663–670.

  36. Tamássy L., Binh T.Q.: On weak symmetries of Einstein and Sasakian manifolds. Tensor, (N.S.), 53, 140–148 (1993)

    MATH  MathSciNet  Google Scholar 

  37. A. G. Walker, On Ruse’s space of recurrent curvature, Proc. London Math. Soc., 52 (1950), 36–54.

  38. K. Yano and M. Kon, Structures on manifolds, World Scientific (Singapore, 1984), 418–421.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata, 700019, West Bengal, India

    U. C. De

  2. Physics Department, Università degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy

    C. A. Mantica

  3. I. I. S. Lagrange, Via L., Modignani 65, 20161, Milano, Italy

    C. A. Mantica

  4. Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea

    Y. J. Suh

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  1. U. C. De
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  2. C. A. Mantica
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  3. Y. J. Suh
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Correspondence to C. A. Mantica.

Additional information

The third author was supported by Proj. No. NRF-2012-R1A2A2A-01043023 from National Research Foundation.

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De, U.C., Mantica, C.A. & Suh, Y.J. On weakly cyclic Z symmetric manifolds. Acta Math. Hungar. 146, 153–167 (2015). https://doi.org/10.1007/s10474-014-0462-9

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  • DOI: https://doi.org/10.1007/s10474-014-0462-9

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