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Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature

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Abstract

In this paper, we show that a generalized Sasakian space form of dimension >3 is either of constant sectional curvature, or a canal hypersurface in Euclidean or Minkowski spaces, or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold, or locally a warped product of a real line and a generalized complex space form, or an \({\alpha}\)-Sasakian space form, or it is of five dimension and admits an \({\alpha}\)-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension >5 is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension >3 is also given in this paper.

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References

  1. Alegre P., Blair D.E., Carriazo A.: Generalized Sasakian-space-forms. Isr. J. Math. 141, 157–183 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alegre P., Carriazo A.: Generalized Sasakian space forms and conformal changes of the metric. Results Math. 59, 485–493 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alegre P., Carriazo A.: Structures on generalized Sasakian-space-forms. Differ. Geom. Appl. 26, 656–666 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair D.E., Vanhecke L.: Symmetries and \({\varphi}\)-symmetric spaces. Tohoku Math. J. 39, 373–383 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyer C.P., Galicki K.: 3-Sasakian manifolds. Surv. Differ. Geom. 6, 123–184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen B.Y., Yano K.: Hypersurfaces of a conformally flat space. Tensor N.S. 26, 318–322 (1972)

    MathSciNet  MATH  Google Scholar 

  7. Fernández-López, M., García-Río, E., Kupeli, D.N., Ünal, B.: A curvature condition for a twisted product to be a warped product. Manuscr. Math. 106, 213–217 (2001)

  8. Falcitelli M.: Locally conformal C 6-manifolds and generalized Sasakian-space-forms. Mediterr. J. Math. 7, 19–36 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Falcitelli M.: A class of almost contact metric manifolds and twisted products. Balk. J. Geom. Appl. 17, 17–29 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Ganchev G., Mihova V.: Riemannian manifolds of quasi-constant sectional curvatures. J. Reine Angew. Math. 522, 119–141 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Ganchev, G., Mihova, V.: A classifications of Riemannian manifolds of quasi-constant sectional curvatures. arXiv:1105.3081

  12. Ghosh A.: Killing vector fields and twistor forms on generalized Sasakian space forms. Mediterr. J. Math. 10, 1051–1065 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kenmotsu K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim U.K.: Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms. Note di Mat. 26, 55–67 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kowalski O., Vanhecke L.: Classification of five-dimensional naturally reductive spaces. Math. Proc. Camb. Philos. 97, 445–463 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kowalski O., Wegrzynowski S.: A classification of five-dimensional \({\varphi}\)-symmetric spaces. Tensor N.S. 46, 379–386 (1987)

    MATH  Google Scholar 

  17. Marrero J.C.: The local structure of trans-Sasakian manifolds. Ann. Mat. Pura Appl. 162, 77–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meumertzheim M., Reckziegel H., Schaaf M.: Decomposition of twisted and warped product nets. Results Math. 36, 297–312 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Okumura M.: Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphic sectional curvatures. Tohoku Math. J. 16, 270–284 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Okumura M.: Some remarks on spaces with certain contact structure. Tohoku Math. J. 14, 135–145 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  21. Olszak Z.: On the existence of generalized complex space forms. Isr. J. Math. 65, 214–218 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. O’Neill, B.: Semi-Riemannian geometry with applications to relatively. In: Pure and Appl. Math. vol. 103. Academic Press, New York (1983)

  23. Schouten, J.: Ricci-Calculus. Springer, Berlin (1954)

  24. Tricerri F., Vanhecke L.: Curvature tensors on almost Hermitian manifolds. Trans. Am. Math. Soc. 267, 365–398 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vanhecke, L.: Almost Hermitian manifolds with J-invariant Riemann curvature tensor. Rend. Sem. Mat. Univ. Politec. Torino 34, 487–498 (1975–1976)

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Authors and Affiliations

  1. Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Jalan Sungai Long, 43000, Cheras, Malaysia

    Avik De

  2. Institute of Mathematical Sciences, University of Malaya, 50603, Kuala Lumpur, Malaysia

    Tee-How Loo

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  1. Avik De
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  2. Tee-How Loo
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Correspondence to Avik De.

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De, A., Loo, TH. Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature. Mediterr. J. Math. 13, 3797–3815 (2016). https://doi.org/10.1007/s00009-016-0715-7

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  • DOI: https://doi.org/10.1007/s00009-016-0715-7

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