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Certain types of metrics on almost coKähler manifolds

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Abstract

In this paper, we study an almost coKähler manifold admitting certain metrics such as \(*\)-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKähler 3-manifold (M, g) admitting a \(*\)-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKähler \((\kappa ,\mu )\)-almost coKähler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a \((\kappa , \mu )\)-almost coKähler manifold (M, g) is coKähler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKähler manifolds which are non-coKähler.

Résumé

Dans cet article, nous étudions une variété presque coKähler admettant certaines métriques telles que les solitons \(*\)-Ricci, satisfaisant l’équation du point critique (CPE) ou Bach-plate. Tout d’abord, nous considérons une 3-variété coKähler (M, g) admettant un \(*\)-Ricci soliton (g, X) et nous montrons dans ce cas que soit M est localement plate ou X est une transformation de contact infinitésimale. Ensuite, nous étudions les métriques non-coKähler \((\kappa , \mu )\)-presque coKähler comme métriques CPE et nous prouvons que g ne peut pas être une solution de CPE avec une fonction non triviale f. Enfin, nous prouvons qu’une variété \((\kappa , \mu )\)-presque coKähler (M, g) est coKähler si M admet un tenseur de Cotton sans divergence ou la métrique g est Bach-plate. En revanche, nous montrons par un exemple approprié qu’il y a des variétés Bach-plates presque coKähler qui ne sont pas coKähler.

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References

  1. Bach, R.: Zur weylschen relativitatstheorie und der Weylschen erweiterung des krummungstensorbegriffs. Math. Z. 9, 110–135 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barros, A., Ribeiro, E., Jr.: Critical point equation on four-dimensional compact manifolds. Math. Nachr. 287, 1618–1623 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Besse, A.: Einstein Manifolds. Springer, New York (2008)

    MATH  Google Scholar 

  4. Blair, D.E.: The theory of quasi-Sasakian structures. J. Differ. Geom. 1, 331–345 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, Vol. 203, Birkhäuser, New York, (2010)

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Case, J., Shu, Y., Wei, G.: Rigidity of quasi-Einstein metrics. Differential Geom. Appl. 29, 93–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Q., He, C.: On Bach flat warped product Einstein manifolds. Pac. J. Math. 265, 313–326 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dacko, P.: On almost cosymplectic manifolds with the structure vector field \(\xi \)-belonging to the \(\kappa \)-nullity distribution. Balkan J. Geom. Appl. 5(2), 47–60 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Dai, X.: Non-existence of \(*\)-Ricci solitons on \((\kappa,\mu )\)-almost cosymplectic manifolds. J. Geom. 110, 30 (2019)

    Article  MathSciNet  Google Scholar 

  11. Dai, X., Zhao, Y., De, U.C.: \(*\)-Ricci soliton on \((\kappa,\mu )^{\prime }\)-almost Kenmotsu manifolds. Open Math. 17, 874–882 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

  13. Ghosh, A., Patra, D.S.: The critical point equation and contact geometry. J. Geom. 108, 185–194 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ghosh, A., Patra, D.S.: \(*\)-Ricci soliton within the frame-work of Sasakian and \((\kappa, \mu )\)-contact manifold. Int. J. Geom. Methods Mod. Phys. 15(7), 1850120 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh, A., Sharma, R.: Sasakian manifolds with purely transversal Bach tensor. J. Math. Phys. 58, 103502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghosh, A., Sharma, R.: Classification of \((\kappa,\mu )\)-contact manifolds with divergence free Cotton tensor and vanishing Bach tensor. Ann. Polon. Math. 122, 153–163 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ghosh, A., Sharma, R.: Bochner-Kähler and Bach flat manifolds. Arch. Math. 113, 551–560 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goldberg, S.I., Yano, K.: Integrability of almost cosymplectic structures. Pacific J. Math. 31, 373–382 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hamada, T.: Real hypersurfaces of complex space forms in terms of Ricci \(*\)-tensor. Tokyo J. Math. 25, 473–483 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)

    Article  MathSciNet  Google Scholar 

  21. Huchchappa, A.K., Naik, D.M., Venkatesha, V.: Certain results on contact metric generalized \((\kappa,\mu )\)-space form. Commun. Korean Math. Soc. 34(4), 1315–1328 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Hwang, S.: Critical points of the total scalar curvature functionals on the space of metrics of constant scalar curvature. Manuscr. Math. 103, 135–142 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kaimakamis, G., Panagiotidou, K.: \(*\)-Ricci solitons of real hypersurface in non-flat complex space forms. J. Geom. Phy. 76, 408–413 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Majhi, P., De, U.C., Suh, Y.J: \(*\)-Ricci solitons on Sasakian 3-Manifolds. Publ. Math. Debrecen. 93, 241–252 (2018)

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

  29. Prakasha, D.G., Veeresha, P.: Para-Sasakian manifolds and \(*\)-Ricci solitons. Afr. Mat. 30, 989–998 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tachibana, S.: On almost-analytic vectors in almost Kahlerian manifolds. Tohoku Math. J. 11, 247–265 (1959)

    MathSciNet  MATH  Google Scholar 

  31. Tanno, S.: The automorphism group of almost contact Riemannian manifolds. Tohoku Math. J. 21, 21–38 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  32. Venkatesha, V., Naik, D.M., Kumara, H.A.: \(*\)-Ricci solitons and gradient almost \(*\)-Ricci solitons on Kenmotsu manifolds. Math. Slovaca 69, 1–12 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Venkatesha, V., Kumara, H.A., Naik, D.M.: Almost \(*\)-Ricci soliton on paraKenmotsu manifolds. Arab. J. Math. (2019). https://doi.org/10.1007/s40065-019-00269-7

    Article  MATH  Google Scholar 

  34. Yano, K.: Integral formulas in Riemannian geometry. Marcel Dekker, New York (1970)

    MATH  Google Scholar 

  35. Yun, G., Hwang, S., Chang, J.: Total scalar curvature and harmonic curvature. Taiwan. J. Math. 18(5), 1439–1458 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Devaraja Mallesha Naik

    Present address: Department of Mathematics, CHRIST (Deemed to be University), Bangalore, Karnataka, India

Authors and Affiliations

  1. Department of Mathematics, Kuvempu University, Shivamogga, Karnataka, India

    Devaraja Mallesha Naik, V. Venkatesha & H. Aruna Kumara

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  1. Devaraja Mallesha Naik
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  2. V. Venkatesha
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  3. H. Aruna Kumara
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Correspondence to V. Venkatesha.

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D.M. Naik is thankful to UGC, New Delhi for financial assistance in the form of SRF. All the authors are thankful to Department of Science and Technology, New Delhi for financial assistance to the Department of Mathematics, Kuvempu University under the FIST program (Ref. No. SR/FST/MS-I/2018/23(C))

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Naik, D.M., Venkatesha, V. & Kumara, H.A. Certain types of metrics on almost coKähler manifolds. Ann. Math. Québec 47, 331–347 (2023). https://doi.org/10.1007/s40316-021-00162-w

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  • DOI: https://doi.org/10.1007/s40316-021-00162-w

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