Advertisement

Almost Contact Metric Manifolds with Local Riemannian and Ricci Symmetries

  • Yaning WangEmail author
Article
  • 67 Downloads

Abstract

In this paper, we show that the Reeb sectional curvature of a locally symmetric almost coKähler manifold \(M^{2n+1}\) is a constant if and only if \(M^{2n+1}\) is locally isometric to the product of \(\mathbb {R}\) and a locally symmetric almost Kähler manifold. Similar result in the framework of almost Kenmotsu manifolds is established. We give a characterization for a Ricci symmetric almost Kenmotsu manifold to be Einstein.

Keywords

Almost coKähler manifold almost Kenmotsu manifold local symmetry Ricci symmetry 

Mathematics Subject Classification

Primary 53D15 Secondary 53C25 

Notes

Acknowledgements

This work was supported by the Youth Science Foundation of Henan Normal University (No. 2014QK01). The author would like to thank the reviewer for his or her useful suggestions.

References

  1. 1.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefGoogle Scholar
  2. 2.
    Blair, D.E.: The theory of quasi-Sasakian structures. J. Differ. Geom. 1, 331–345 (1967)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. In: Progress in Mathematics, vol. 203. Birkhäuser, Basel (2010)CrossRefGoogle Scholar
  4. 4.
    Blair, D.E., Goldberg, S.I.: Topology of almost contact manifolds. J. Differ. Geom. 1, 347–354 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cappelletti-Montano, B., De Nicola, A., Yudin, I.: A survey on cosymplectic geometry, Rev. Math. Phys. 25, 1343002, 55 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cho, J.T.: Local symmetry on almost Kenmotsu three-manifolds. Hokkaido Math. J. 45, 435–442 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cho, J.T.: Reeb flow symmetry on almost cosymplectic three-manifolds. Bull. Korean Math. Soc. 53, 1249–1257 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dacko, P.: On almost cosymplectic manifolds with the structure vector field \(\xi \) belonging to the \(k\)-nullity distribution. Balkan J. Geom. Appl. 5, 47–60 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14, 343–354 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93, 46–61 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Goldberg, S.I., Yano, K.: Integrability of almost cosymplectic structures. Pac. J. Math. 31, 373–382 (1969)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tôhoku Math. J. 24, 93–103 (1972)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kirichenko, V.F., Kharitonova, S.V.: On the geometry of normal locally conformal almost cosymplectic manifolds. Math. Notes 91, 34–45 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Olszak, Z.: On almost cosymplectic manifolds. Kodai Math. J. 4, 239–250 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Olszak, Z.: Almost cosymplectic manifolds with Kählerian leaves. Tensor (N. S.) 46, 117–124 (1987)zbMATHGoogle Scholar
  17. 17.
    Olszak, Z., Ro̧sca, R.: Normal locally conformal almost cosymplectic manifolds. Publ. Math. Debr. 39, 315–323 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pastore, A.M., Saltarelli, V.: Almost Kenmotsu manifolds with conformal Reeb foliation. Bull. Belg. Math. Soc. Simon Stevin 18, 655–666 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds. Differ. Geom. Appl. 30, 49–58 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, Y., Liu, X.: Locally symmetric \(CR\)-integrable almost Kenmotsu manifolds. Mediterr. J. Math. 12, 159–171 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, Y.: Three-dimensional locally symmetric almost Kenmotsu manifolds. Ann. Polon. Math. 116, 79–86 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Wang, Y.: Almost coKähler manifolds satisfying some symmetry conditions. Turk. J. Math. 40, 740–752 (2016)CrossRefGoogle Scholar
  23. 23.
    Wang, Y.: Locally symmetric almost coKähler \(5\)-manifolds with Kählerian leaves. Bull. Korean Math. Soc. 55, 789–798 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information SciencesHenan Normal UniversityXinxiangPeople’s Republic of China

Personalised recommendations