Research Article
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Characterization of a paraSasakian manifold admitting Bach tensor

Year 2023, Volume: 72 Issue: 3, 826 - 838, 30.09.2023

U.c. De Gopal Ghosh Krishnendu De

https://doi.org/10.31801/cfsuasmas.1172289

Abstract

In the present article, our aim is to characterize Bach flat paraSasakian manifolds. It is established that a Bach flat paraSasakian manifold of dimension greater than three is of constant scalar curvature. Next, we prove that if the metric of a Bach flat paraSasakian manifold is a Yamabe soliton, then the soliton field becomes a Killing vector field. Finally, it is shown that a 3-dimensional Bach flat paraSasakian manifold is locally isometric to the hyperbolic space $H^{2n+1}(1)$.

Keywords

Bach tensor cotton tensor paraSasakian manifold

References

  • Adati, T., Matsumuto. K., On conformally recurrent and conformally symmetric P-Sasakianmanifolds, TRU Math., 13 (1977), 25-32.
  • Bach, R., Zur weylschen relativitatstheorie und der Weylschen erweiterung des krummungstensorbegriffs, Math. Z., 9 (1921), 110-135.
  • Bergman, J., Conformal Einstein spaces and Bach tensor generalization in n-dimensions, Thesis, Linkoping (2004).
  • Deshmukh, S., Chen, B. Y., A note on Yamabe solitons, Balk. J. Geom. Appl., 23 (2018), 37-43.
  • De, K., De, U. C., A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds, Korean J. Math., 28 (2020), 739-751. https://doi.org/10.11568/kjm.2020.28.4.739
  • De, K., De, U. C., δ-almost Yamabe solitons in paracontact metric manifolds, Mediterr. J. Math., 18, 218 (2021). DOI:10.1007/s00009-021-01856-9
  • Duggal, K. L., Sharma, R., Symmetries of spacetimes and Riemannian manifolds, Kluwer Academic Publishers, 1999.
  • De, U. C., Ghosh, G., Jun, J. B., Majhi, P., Some results on paraSasakian manifolds, Bull. Translivania Univ. Brasov, Series III:Mathematics, Informatics, Physics., 60 (2018), 49-64.
  • Erken, I. K., Dacko, P. and Murathan, C., Almost paracosymplectic manifolds, J. Geom. Phys., 88 (2015), 30-51. DOI: 10.21099/tkbjm/1496164721
  • Erken, I. K. and Murathan, C., A complete study of three-dimensional paracontact $(\widetilde{\kappa},\widetilde{\mu},\widetilde{\nu})$-spaces, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750106.https://doi.org/10.1142/S0219887817501067
  • Erken, I. K., Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Periodica Mathematica Hungarica, 80 (2020), 172-184. https://doi.org/10.1007/s10998-019-00303-3
  • Fu, H. P. and Peng, J. K., Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature, Hokkaido Math. J., 47 (2018), 581-605. https://doi.org/10.14492/HOKMJ/2F1537948832
  • Hamilton, R., The Ricci flow on surface, Contemp. Math., 71 (1988), 237-267.
  • Pedersen, H., Swann, A., Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew. Math., 441 (1993), 99-113.
  • Sato, I., On a structure similar to the almost contact structure, Tensor, N. S., 30 (1976), 219-224.
  • Suh, Y. J., De, U. C., Yamabe soliton and Ricci solitons on almost co-Kahler manifolds, Can. Math. Bull., 62 (2019), 653-661. http://dx.doi.org/10.4153/S0008439518000693
  • Matsumuto. K., Ianus, S. and Mihai, I., On P-Sasakian manifolds which admit certain tensor fields, Publ. Math. Debrecen., 33 (1986), 61-65.
  • Mihai, I., Rosca, R., On Lorentzian P-Sasakian Manifolds, Classical Analysis,World Scientific Publ., 156-169 (1992).
  • Mihai, I., Some structures defined on the tangent bundle of a P-Sasakian manifold, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N. S), 77 (1985), 61-67.
  • Kaneyuki, S., Willams, F. L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173-187.
  • Kaneyuki, S., Kozai, M., Paracomplex structures and affine symmetric spaces, Tokyo J. of Math., 08 (1985), 81-98.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to the Relativity, Academic Press, New York-London, 1983.
  • Sharma, R., Ghosh, A., Sasakian manifolds with purely transversal Bach tensor, J. Math. Phys., 58 103502 (2017). doi: 10.1063/1.4986492
  • Sharma, R., Ghosh, A., Classification of $(k,\mu)$-contact manifolds with divergence free Cotton tensor and Vanishing Bach tensor, Ann. Polon. Math., (2019). DOI: 10.4064/ap 180228-13-11
  • Szekeres, P., Conformal tensors, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 304, No. 1476 (Apr. 2, 1968), 113-122.
  • Wang, Y., Yamabe solitons on three dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc., 23 (2016), 345-355. http://dx.doi.org/10.36045/bbms/1473186509
  • Zamkovoy, S., Canonical connection on paracontact manifolds, Ann. Glob. Anal. Geom., 36 (2009), 37-60. http://dx.doi.org/10.1007/s10455-008-9147-3

Year 2023, Volume: 72 Issue: 3, 826 - 838, 30.09.2023

U.c. De Gopal Ghosh Krishnendu De

https://doi.org/10.31801/cfsuasmas.1172289

Abstract

References

  • Adati, T., Matsumuto. K., On conformally recurrent and conformally symmetric P-Sasakianmanifolds, TRU Math., 13 (1977), 25-32.
  • Bach, R., Zur weylschen relativitatstheorie und der Weylschen erweiterung des krummungstensorbegriffs, Math. Z., 9 (1921), 110-135.
  • Bergman, J., Conformal Einstein spaces and Bach tensor generalization in n-dimensions, Thesis, Linkoping (2004).
  • Deshmukh, S., Chen, B. Y., A note on Yamabe solitons, Balk. J. Geom. Appl., 23 (2018), 37-43.
  • De, K., De, U. C., A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds, Korean J. Math., 28 (2020), 739-751. https://doi.org/10.11568/kjm.2020.28.4.739
  • De, K., De, U. C., δ-almost Yamabe solitons in paracontact metric manifolds, Mediterr. J. Math., 18, 218 (2021). DOI:10.1007/s00009-021-01856-9
  • Duggal, K. L., Sharma, R., Symmetries of spacetimes and Riemannian manifolds, Kluwer Academic Publishers, 1999.
  • De, U. C., Ghosh, G., Jun, J. B., Majhi, P., Some results on paraSasakian manifolds, Bull. Translivania Univ. Brasov, Series III:Mathematics, Informatics, Physics., 60 (2018), 49-64.
  • Erken, I. K., Dacko, P. and Murathan, C., Almost paracosymplectic manifolds, J. Geom. Phys., 88 (2015), 30-51. DOI: 10.21099/tkbjm/1496164721
  • Erken, I. K. and Murathan, C., A complete study of three-dimensional paracontact $(\widetilde{\kappa},\widetilde{\mu},\widetilde{\nu})$-spaces, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750106.https://doi.org/10.1142/S0219887817501067
  • Erken, I. K., Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Periodica Mathematica Hungarica, 80 (2020), 172-184. https://doi.org/10.1007/s10998-019-00303-3
  • Fu, H. P. and Peng, J. K., Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature, Hokkaido Math. J., 47 (2018), 581-605. https://doi.org/10.14492/HOKMJ/2F1537948832
  • Hamilton, R., The Ricci flow on surface, Contemp. Math., 71 (1988), 237-267.
  • Pedersen, H., Swann, A., Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew. Math., 441 (1993), 99-113.
  • Sato, I., On a structure similar to the almost contact structure, Tensor, N. S., 30 (1976), 219-224.
  • Suh, Y. J., De, U. C., Yamabe soliton and Ricci solitons on almost co-Kahler manifolds, Can. Math. Bull., 62 (2019), 653-661. http://dx.doi.org/10.4153/S0008439518000693
  • Matsumuto. K., Ianus, S. and Mihai, I., On P-Sasakian manifolds which admit certain tensor fields, Publ. Math. Debrecen., 33 (1986), 61-65.
  • Mihai, I., Rosca, R., On Lorentzian P-Sasakian Manifolds, Classical Analysis,World Scientific Publ., 156-169 (1992).
  • Mihai, I., Some structures defined on the tangent bundle of a P-Sasakian manifold, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N. S), 77 (1985), 61-67.
  • Kaneyuki, S., Willams, F. L., Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173-187.
  • Kaneyuki, S., Kozai, M., Paracomplex structures and affine symmetric spaces, Tokyo J. of Math., 08 (1985), 81-98.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to the Relativity, Academic Press, New York-London, 1983.
  • Sharma, R., Ghosh, A., Sasakian manifolds with purely transversal Bach tensor, J. Math. Phys., 58 103502 (2017). doi: 10.1063/1.4986492
  • Sharma, R., Ghosh, A., Classification of $(k,\mu)$-contact manifolds with divergence free Cotton tensor and Vanishing Bach tensor, Ann. Polon. Math., (2019). DOI: 10.4064/ap 180228-13-11
  • Szekeres, P., Conformal tensors, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 304, No. 1476 (Apr. 2, 1968), 113-122.
  • Wang, Y., Yamabe solitons on three dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc., 23 (2016), 345-355. http://dx.doi.org/10.36045/bbms/1473186509
  • Zamkovoy, S., Canonical connection on paracontact manifolds, Ann. Glob. Anal. Geom., 36 (2009), 37-60. http://dx.doi.org/10.1007/s10455-008-9147-3
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

U.c. De 0000-0002-8990-4609

Gopal Ghosh 0000-0001-6178-6340

Krishnendu De 0000-0001-6520-4520

Publication Date September 30, 2023
Submission Date September 7, 2022
Acceptance Date May 14, 2023
Published in Issue Year 2023 Volume: 72 Issue: 3

Cite

APA De, U., Ghosh, G., & De, K. (2023). Characterization of a paraSasakian manifold admitting Bach tensor. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(3), 826-838. https://doi.org/10.31801/cfsuasmas.1172289
AMA De U, Ghosh G, De K. Characterization of a paraSasakian manifold admitting Bach tensor. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2023;72(3):826-838. doi:10.31801/cfsuasmas.1172289
Chicago De, U.c., Gopal Ghosh, and Krishnendu De. “Characterization of a ParaSasakian Manifold Admitting Bach Tensor”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 3 (September 2023): 826-38. https://doi.org/10.31801/cfsuasmas.1172289.
EndNote De U, Ghosh G, De K (September 1, 2023) Characterization of a paraSasakian manifold admitting Bach tensor. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 3 826–838.
IEEE U. De, G. Ghosh, and K. De, “Characterization of a paraSasakian manifold admitting Bach tensor”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 3, pp. 826–838, 2023, doi: 10.31801/cfsuasmas.1172289.
ISNAD De, U.c. et al. “Characterization of a ParaSasakian Manifold Admitting Bach Tensor”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/3 (September 2023), 826-838. https://doi.org/10.31801/cfsuasmas.1172289.
JAMA De U, Ghosh G, De K. Characterization of a paraSasakian manifold admitting Bach tensor. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:826–838.
MLA De, U.c. et al. “Characterization of a ParaSasakian Manifold Admitting Bach Tensor”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 3, 2023, pp. 826-38, doi:10.31801/cfsuasmas.1172289.
Vancouver De U, Ghosh G, De K. Characterization of a paraSasakian manifold admitting Bach tensor. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(3):826-38.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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