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Gradient Yamabe and Gradient m-Quasi Einstein Metrics on Three-dimensional Cosymplectic Manifolds

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Abstract

In this paper, we characterize the gradient Yamabe and the gradient m-quasi Einstein solitons within the framework of three-dimensional cosymplectic manifolds.

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References

  1. Ayar, G., Chaubey, S.K.: \(M\)-projective curvature tensor over cosymplectic manifolds. Differ. Geom. Dyn. Syst. 21, 23–33 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Barros, A., Gomes, J.N.: A compact gradient generalized quasi-Einstein metric with constant scalar curvature. J. Math. Anal. Appl. 401, 702–705 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barros, A., Ribeiro Jr., E: Some characterizations for compact almost Ricci solitons. Proc. Am. Math. Soc. 140, 1033–1040 (2012)

  4. Blair, D.E.: Contact manifold in Riemannian geometry. Lecture Notes in Mathematics, vol. 509. Springer-Verlag, Berlin (1976)

  5. Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. Progress Math. 203, Birkhauser (2010)

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

    MathSciNet  MATH  Google Scholar 

  7. Boothby, M.M., Wang, H.C.: On contact manifolds. Ann. Math. 68, 421–450 (1958)

    Article  MATH  Google Scholar 

  8. Cao, H.D.: Recent progress on Ricci soliton. Adv. Lect. Math. 11, 1–38 (2009)

    MathSciNet  Google Scholar 

  9. Cao, X.: Compact gradient shrinking Ricci solitons with positive curvature operator. J. Geom. Anal. 17, 425–433 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Carriazo, A., Martin-Molina, V.: Almost cosymplectic and almost Kenmotsu \((\kappa, \mu, \nu )\)-spaces. Mediterr. J. Math. 10, 1551–1571 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Case, J.: On the nonexistence of quasi-Einstein metrics. Pac. J. Math. 248, 227–284 (2010)

    Article  MathSciNet  Google Scholar 

  13. Case, J., Shu, Y., Wei, G.: Rigidity of quasi-Einstein metrics. Differ. Geom. Appl. 29, 93–100 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chaubey, S.K., De, U.C.: Characterization of three-dimensional Riemannian manifolds with a type of semi-symmetric metric connection admitting Yamabe soliton. J. Geom. Phys. (2020). https://doi.org/10.1016/j.geomphys.2020.103846

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

    MathSciNet  MATH  Google Scholar 

  16. Cho, J.T.: Reeb flow symmetry on almost cosymplectic three-manifold. Bull. Korean Math. Soc. 53, 1249–1257 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cho, J.T.: Conformally flat normal almost contact 3-manifolds. Honam Math. J. 38, 59–69 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Dacko, P., Olszak, Z.: On conformally flat almost cosymplectic manifolds with K\(\ddot{a}\)hlerian leaves. Rend. Sem. Mat. Univ. Pol. Torino 56, 89–103 (1998)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Dacko, P., Olszak, Z.: On almost cosymplectic \((\kappa , \mu , \nu )\)-spaces. PDEs, submanifolds and affine differential geometry. 211–220, Banach Center Publ. 69, Polish Acad. Sci. Inst. Math., Warsaw (2005)

  23. De, U.C., Chaubey, S.K., Suh, Y.J.: A note on almost co-Kähler manifolds. Int. J. Geom. Methods Mod. Phys. (2020). https://doi.org/10.1142/S0219887820501534

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

  25. Erken, I.K., Murathan, C.: A class of \(3\)-dimensional almost cosymplectic manifolds. Turk. J. Math. 37, 884–894 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Goldberg, S.I., Yano, K.: Integrability of almost cosymplectic structures. Pacific J. Math. 31, 373–38 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ghosh, A.: Generalized \(m\)-quasi-Einstein metric within the framework of Sasakian and \(K\)-contact manifolds. Ann. Polonici Math. 115, 33–41 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ghosh, A.: \(m\)-quasi-Einstein metric and contact geometry. RACSAM 113, 2587–2600 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hamilton, R.S.: The Ricci flow on surfaces. Math. Gen. Relativ. (Santa Cruz, CA, 1986), Contemp. Math. 71, 237–262 (1998)

  30. He, C., Petersen, P., Wylie, W.: On the classification of warped product Einstein metrics. Comm. Anal. Geom. 20, 271–312 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hu, Z., Li, D., Xu, J.: On generalized m-quasi-Einstein manifolds with constant scalar curvature. J. Math. Anal. Appl. 432, 733–743 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, H.: Topology of co-symplectic/co-K\(\ddot{a}\)hler manifolds. Asian J. Math. 12, 527–544 (2008)

    Article  MathSciNet  Google Scholar 

  33. Ma, L., Cheng, L.: Properties of complete non-compact Yamabe solitons. Ann. Glob. Anal. Geom. 40, 379–387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. Oztürk, H., Aktan, N., Murathan, C.: Almost \(\alpha \)-cosymplectic \((\kappa , \mu , \nu )\)-spaces. (2010) arxiv:1007.0527v1

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

    Article  MathSciNet  MATH  Google Scholar 

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

  38. Olszak, Z.: Normal almost contact metric manifolds of dimension three. Ann. Polon. Math. 47, 41–50 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Olszak, Z.: Locally conformal almost cosymplectic manifolds. Colloq. Math. 57, 73–87 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint. (2002) arXiv:math.DG/02111159

  41. Perrone, D.: Classification of homogeneous almost cosymplectic three-manifolds. Differ. Geom. Appl. 30, 49–58 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Perrone, D.: Minimal Reeb vector fields on almost cosymplectic manifolds. Kodai Math. J. 36, 258–274 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sasaki, S.: On differentiable manifolds with certain structures which are closely related to almost contact structure \(I\). Tohoku Math. J. 2(12), 459–476 (1960)

    MathSciNet  MATH  Google Scholar 

  44. Suh, Y.J., De, U.C.: Yamabe solitons and Ricci solitons on almost co-Kähler manifolds. Canad. Math. Bull. 62, 1–9 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tanno, S.: Note on infinitesimal transformations over contact manifolds. Tohoku Math. J. 14, 416–430 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  46. Turan, M., Yetim, C., Chaubey, S.K.: On quasi-Sasakian \(3\)-manifolds admitting \(\eta \)-Ricci solitons. Filomat 33(15), 4923–4930 (2019)

    Article  MathSciNet  Google Scholar 

  47. Wang, L.F.: On non compact quasi Yamabe gradient solitons. Differ. Geom. Appl. 31, 337–347 (2013)

    Article  MATH  Google Scholar 

  48. Wang, W.: A class of three dimensional almost co-Kähler manifolds. Palestine J. Math. 6, 111–118 (2017)

    MathSciNet  Google Scholar 

  49. Wang, Y.: A generalization of Goldberg conjecture for co-Kähler manifolds. Mediterr. J. Math. 13, 2679–2690 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, Y.: Ricci tensors on three-dimensional almost co-Kähler manifolds. Kodai Math. J. 39, 469–483 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, Y.: Ricci solitons on \(3\)-dimensional cosymplectic manifolds. Math. Slovaca 67, 979–984 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wei, G., Wylie, W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Differ. Geom. 83, 337–405 (2009)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors express their sincere thanks to the anonymous referees for providing the valuable suggestions in the improvement of the paper. The third author was supported by Grant Proj. No. NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea. The second author acknowledges authority of University of Technology and Applied Sciences-Shinas, Oman for their continuous support and encouragement to carry out this research work.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata, West Bengal, 700019, India

    Uday Chand De

  2. Section of Mathematics, Department of Information Technology, University of Technology and Applied Sciences-Shinas, P.O. Box 77, Shinas, Postal Code 324, Oman

    Sudhakar K. Chaubey

  3. Department of Mathematics and RIRCM, Kyungpook National University, Daegu, 41566, South Korea

    Young Jin Suh

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  1. Uday Chand De
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  2. Sudhakar K. Chaubey
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  3. Young Jin Suh
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Correspondence to Sudhakar K. Chaubey.

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De, U.C., Chaubey, S.K. & Suh, Y.J. Gradient Yamabe and Gradient m-Quasi Einstein Metrics on Three-dimensional Cosymplectic Manifolds. Mediterr. J. Math. 18, 80 (2021). https://doi.org/10.1007/s00009-021-01720-w

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