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BACH ALMOST SOLITONS IN PARASASAKIAN GEOMETRY

  • Uday Chand De (Department of Pure Mathematics University of Calcutta) ;
  • Gopal Ghosh (Department of Basic Science and Humanities Cooch Behar Government Engineering College)
  • Received : 2022.05.28
  • Accepted : 2022.11.22
  • Published : 2023.05.31
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Abstract

If a paraSasakian manifold of dimension (2n + 1) represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric g has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if r = -6; shrinking if r > -6; expanding if r < -6.

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Acknowledgement

The authors are grateful to the referee for his or her valuable suggestions and comments which improved the quality of this paper.

References

  1. T. Adati and K. Matsumoto, On conformally recurrent and conformally symmetric P-Sasakian manifolds, TRU Math. 13 (1977), no. 1, 25-32. 
  2. R. Bach, Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungstensorbegriffs, Math. Z. 9 (1921), no. 1-2, 110-135. https://doi.org/10.1007/BF01378338 
  3. E. Bahuaud and D. Helliwell, Short-time existence for some higher-order geometric flows, Comm. Partial Differential Equations 36 (2011), no. 12, 2189-2207. https://doi.org/10.1080/03605302.2011.593015 
  4. I. Bakas, F. Bourliot, D. Lust, and M. Petropoulos, Geometric flows in Horava-Lifshitz gravity, J. High Energy Phys. 2010 (2010), no. 4, 131, 58 pp. https://doi.org/10.1007/JHEP04(2010)131 
  5. J. Bergman, Conformal Einstein spaces and Bach tensor generalization in n-dimensions, Thesis, Linkoping, 2004. 
  6. A. M. Blaga, Conformal and paracontactly geodesic transformations of almost paracontact metric structures, Facta Univ. Ser. Math. Inform. 35 (2020), no. 1, 121-130.  https://doi.org/10.22190/FUMI2001121B
  7. A. M. Blaga and M. Crasmareanu, Statistical structures in almost paracontact geometry, Bull. Iranian Math. Soc. 44 (2018), no. 6, 1407-1413. https://doi.org/10.1007/s41980-018-0088-8 
  8. G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math. 55 (2011), no. 2, 697-718. http://projecteuclid.org/euclid.ijm/1359762409  https://doi.org/10.1215/ijm/1359762409
  9. G. Calvaruso and A. Perrone, Ricci solitons in three-dimensional paracontact geometry, J. Geom. Phys. 98 (2015), 1-12. https://doi.org/10.1016/j.geomphys.2015.07.021 
  10. H.-D. Cao and Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149-1169. https://doi.org/10.1215/00127094-2147649 
  11. S. Das and S. Kar, Bach flows of product manifolds, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 5, 1250039, 18 pp. https://doi.org/10.1142/S0219887812500399 
  12. U. C. De, G. Ghosh, and K. De, A note on Bach flat paraSasakian manifold, Communicated. 
  13. U. C. De, G. Ghosh, J. B. Jun, and P. Majhi, Some results on paraSasakian manifolds, Bull. Transilv. Univ. Bra,sov Ser. III 11(60) (2018), no. 1, 49-63. 
  14. U. C. De and A. Sardar, Classification of (k, µ)-almost co-Kahler manifolds with vanishing Bach tensor and divergence free Cotton tensor, Commun. Korean Math. Soc. 35 (2020), no. 4, 1245-1254. https://doi.org/10.4134/CKMS.c200091 
  15. U. C. De and A. Sarkar, On a type of P-Sasakian manifolds, Math. Rep. (Bucur.) 11(61) (2009), no. 2, 139-144. 
  16. H. Fu and J. Peng, Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature, Hokkaido Math. J. 47 (2018), no. 3, 581-605. https://doi.org/10.14492/hokmj/1537948832 
  17. A. Ghosh, On Bach almost solitons, Beitr. Algebra Geom. 63 (2022), no. 1, 45-54. https://doi.org/10.1007/s13366-021-00565-4 
  18. A. Ghosh and R. Sharma, Sasakian manifolds with purely transversal Bach tensor, J. Math. Phys. 58 (2017), no. 10, 103502, 6 pp. https://doi.org/10.1063/1.4986492 
  19. D. Helliwell, Bach flow on homogeneous products, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 027, 35 pp. https://doi.org/10.3842/SIGMA.2020.027 
  20. P. T. Ho, Bach flow, J. Geom. Phys. 133 (2018), 1-9. https://doi.org/10.1016/j.geomphys.2018.07.008 
  21. S. Kaneyuki and M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985), no. 1, 81-98. https://doi.org/10.3836/tjm/1270151571 
  22. S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173-187. https://doi.org/10.1017/S0027763000021565 
  23. I. Kupeli Erken, Some classes of 3-dimensional normal almost paracontact metric manifolds, Honam Math. J. 37 (2015), no. 4, 457-468. https://doi.org/10.5831/HMJ.2015.37.4.457 
  24. I. Kupeli Erken, Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Period. Math. Hungar. 80 (2020), no. 2, 172-184. https://doi.org/10.1007/s10998-019-00303-3 
  25. V. Martin-Molina, Local classification and examples of an important class of paracontact metric manifolds, Filomat 29 (2015), no. 3, 507-515. https://doi.org/10.2298/FIL1503507M 
  26. I. Sato, On a structure similar to the almost contact structure, Tensor (N.S.) 30 (1976), no. 3, 219-224. 
  27. P. Szekeres, Conformal Tensors, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 304, No. 1476 (Apr. 2, 1968), pp. 113-122.  https://doi.org/10.1098/rspa.1968.0076
  28. Y. Wang, Cotton tensors on almost coKahler 3-manifolds, Ann. Polon. Math. 120 (2017), no. 2, 135-148. https://doi.org/10.4064/ap170410-3-10 
  29. S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom. 36 (2009), no. 1, 37-60. https://doi.org/10.1007/s10455-008-9147-3 
  30. S. Zamkovoy and V. Tzanov, Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform. 100 (2011), 27-34.