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The structure of some classes of K-contact manifolds

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Abstract

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S 2n+1 (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S 1-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.

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References

  1. Blair D E, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203 (Boston, MA: Birkhauser Boston, Inc.) (2002)

    MATH  Google Scholar 

  2. Boothby W M and Wang H C, On contact manifolds, Ann. Math. 68 (1958) 721–734

    Article  MathSciNet  Google Scholar 

  3. Cabrerizo J L, Fernández L M, Fernández M and Zhen G, The structure of a class of K-contact manifolds, Acta Math. Hungar. 82(4) (1999) 331–340

    Article  MATH  MathSciNet  Google Scholar 

  4. Mishra R S, On Sasakian manifolds, Indian J. Pure Appl. Math. 1 (1970) 98–105

    MATH  MathSciNet  Google Scholar 

  5. Ogiue K, On fibrings of almost contact manifolds, Kodai Math. Sem. Rep. 17 (1965) 53–62

    Article  MATH  MathSciNet  Google Scholar 

  6. Ozgur C, φ-conformally flat Lorentzian para-Sasakian manifolds, Rad. Mat. 12(1) (2003) 99–106

    MathSciNet  Google Scholar 

  7. Yano K and Bochner S, Curvature and Betti numbers, Annals of Mathematics Studies 32 (Princeton University Press) (1953)

  8. Zhen G, On conformal symmetric K-contact manifolds, Chinese Quart. J. Math. 7 (1992) 5–10

    MATH  MathSciNet  Google Scholar 

  9. Zhen G, Cabrerizo J L, Fernández L M and Fernández M, On ξ-conformally flat contact metric manifolds, Indian J. Pure Appl. Math. 28 (1997) 725–734

    MATH  MathSciNet  Google Scholar 

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Author notes

  1. Mukut Mani Tripathi

    Present address: Department of Mathematics, Banaras Hindu University, Varanasi, 221 005, India

Authors and Affiliations

  1. Department of Mathematics and Astronomy, Lucknow University, Lucknow, 226 007, India

    Mukut Mani Tripathi & Mohit Kumar Dwivedi

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  1. Mukut Mani Tripathi
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  2. Mohit Kumar Dwivedi
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Correspondence to Mukut Mani Tripathi.

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Tripathi, M.M., Dwivedi, M.K. The structure of some classes of K-contact manifolds. Proc Math Sci 118, 371–379 (2008). https://doi.org/10.1007/s12044-008-0029-1

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  • DOI: https://doi.org/10.1007/s12044-008-0029-1

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