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A Generalization of the Goldberg Conjecture for CoKähler Manifolds

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Abstract

Let M be a compact almost coKähler manifold. If the metric g of M is a Ricci soliton and the potential vector field is pointwise collinear with the Reeb vector field, then we prove that M is Ricci-flat and coKähler and the soliton g is steady. This generalizes a Goldberg-like conjecture for coKähler manifolds obtained by Cappelletti-Montano and Pastore, namely any compact Einstein K-almost coKähler manifold is coKähler. Without the assumption of compactness, Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKähler manifolds are also studied. Moreover, we prove that there exist no gradient Ricci solitons on proper \({(\kappa, \mu)}\)-almost coKähler manifolds.

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References

  1. Apostolov, V., Drăghici, T., Moroianu, A.: The odd dimensional Goldberg conjecture. Math. Nachr. 279(9–10), 948–952 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. In: Progress in Mathematics, vol. 203. Birkhäuser, Boston (2010)

  3. Boyer, C.P., Galicki, K.: Einstein manifolds and contact geometry. Proc. Am. Math. Soc. 129(8), 2419–2430 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cappelletti-Montano, B., Nicola, A.D., Yudin, I.: A survey on cosymplectic geometry. Rev. Math. Phys. 25(10), 1343002 (2013) (55 pages)

  5. Cappelletti-Montano, B., Pastore, A.M.: Einstein-like conditions and cosymplectic geometry. J. Adv. Math. Stud. 3(2), 27–40 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Carriazo, A., Martín-Molina, V.: Almost cosymplectic and almost Kenmotsu \({(k, \mu, \nu)}\)-spaces. Mediterr. J. Math. 10(3), 1551–1571 (2013)

  7. Chinea, D., de León, M., Marrero, J.C.: Topology of cosymplectic manifolds. J. Math. Pures Appl. 72(6), 567–591 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Cho, J.T.: Almost contact 3-manifolds and Ricci solitons. Int. J. Geom. Methods Mod. Phys. 10(1), 1220022 (2013) (7 pages)

  9. Cho, J.T., Sharma, R.: Contact geometry and Ricci solitons. Int. J. Geom. Methods Mod. Phys. 7(6), 951–960 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dacko, P., Olszak, Z.: On almost cosymplectic \({(k,\mu, \nu)}\)-space. Banach Center Publ. 69, 211–220 (2005)

  11. Endo, H.: Non-existence of almost cosymplectic manifolds satisfying a certain condition. Tensor (N. S.) 63(3), 272–284 (2002)

  12. Ghosh, A.: Ricci solitons and contact metric manifolds. Glasgow Math. J. 55(1), 123–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghosh, A., Sharma, R., Cho, J.T.: Contact metric manifolds with \({\eta}\)-parallel torsion tensor. Ann. Glob. Anal. Geom. 34(3), 287–299 (2008)

  14. Goldberg, S.I.: Integrability of almost Kaehler manifolds. Proc. Am. Math. Soc. 21, 96–100 (1969)

  15. Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

  16. Li, H.: Topology of co-symplectic/co-Kähler manifolds. Asian J. Math. 12(4), 527–544 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marrero, J.C., Padrón, E.: New examples of compact cosymplectic solvmanifolds. Arch. Math. (Brno) 34(3), 337–345 (1998)

  18. Nurowski, P., Przanowski, M.: A four-dimensional example of Ricci flat metric admitting almost Kähler non-Kähler structure. Class. Quantum Grav. 16(3), L9–L13 (1999)

  19. Olszak, Z.: On almost cosymplectic manifolds. Kodai Math. J. 4(2), 239–250 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Olszak, Z.: On almost cosymplectic manifolds with Kählerian leaves. Tensor (N. S.) 46, 117–124 (1987)

  21. Oztürk, H., Aktan N., Murathan, C.: Almost \({\alpha}\)-cosymplectic \({(k,\mu,\nu)}\)-spaces. arXiv:1007.0527v1 (2010) (e-prints)

  22. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 (2002) (e-prints)

  23. Sato, T.: Some examples of almost Kähler 4-manifolds. Balkan J. Geom. Appl. 5(2), 113–137 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Sekigawa, K.: On some compact Einstein almost Kahler manifolds. J. Math. Soc. Jpn. 39(4), 677–684 (1987)

  25. Sharma, R.: Certain results on K-contact and \({(k, \mu)}\)-contact manifolds. J. Geom. 89(1–2), 138–147 (2008)

  26. Wang, Y.: Ricci solitons on 3-dimensional cosymplectic manifolds. Math. Slovaca (accepted)

  27. Wang, Y., De, U.C., Liu, X.: Gradient Ricci solitons on almost Kenmotsu manifolds. Publ. Inst. Math. doi:10.2298/PIM140527003W

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  1. Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Sciences, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Yaning Wang

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  1. Yaning Wang
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Wang, Y. A Generalization of the Goldberg Conjecture for CoKähler Manifolds. Mediterr. J. Math. 13, 2679–2690 (2016). https://doi.org/10.1007/s00009-015-0646-8

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  • DOI: https://doi.org/10.1007/s00009-015-0646-8

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