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$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure

Year 2024, , 83 - 92, 23.05.2024

Abhishek Singh , S. K. Chaubey , Sunil Yadav , Shraddha Patel

https://doi.org/10.32323/ujma.1418496

Abstract

The main goal of this manuscript is to investigate the properties of $N(k)$-contact metric manifolds admitting a $\mathcal{Z^\ast}$-tensor. We prove the necessary conditions for which $N(k)$-contact metric manifolds endowed with a $\mathcal{Z^\ast}$-tensor are Einstein manifolds. In this sequel, we accomplish that an $N(k)$-contact metric manifold endowed with a $\mathcal{Z^\ast}$-tensor satisfying $\mathcal{Z^\ast}(\mathcal{G}_{1},\hat{\zeta})\cdot \mathcal{\overset{\star}R}=0$ is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ or an Einstein manifold. We also prove the condition for which an $N(k)$-contact metric manifold endowed with a $\mathcal{Z^\ast}$-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an $N(k)$-contact metric manifold.

Keywords

$N(k)$-contact metric manifold Einstein manifold Ricci soliton $\mathcal{Z}^\star$-recurrent $\mathcal{Z^\ast}$-tensor

References

  • [1] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441–448.
  • [2] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Isr. J. Math., 91 (1995), 189–214.
  • [3] U. C. De, Y. J. Suh, S. K. Chaubey, Conformal vector fields on almost co-K¨ahler manifolds, Math. Slovaca, 71(6) (2021), 1545–1552.
  • [4] U. C. De, S. K. Chaubey, Y. J. Suh, A note on almost co-K¨ahler manifolds, Int. J. Geom. Methods Mod. Phys., 17(10) (2020), 2050153, 14 pp.
  • [5] S. K. Chaubey, M. A. Khan, A. S. R. Al Kaabi, N(k)-paracontact metric manifolds admitting the Fischer-Marsden conjecture, AIMS Math., 9(1) (2024), 2232–2243.
  • [6] S. K. Chaubey, K. K. Bhaishya, M. D. Siddiqi, Existence of some classes of N(k)-quasi Einstein manifolds, Bol. Soc. Parana. Mat., 39(5) (2021), 145–162.
  • [7] S. K. Chaubey, Certain results on N(k)-quasi Einstein manifolds, Afr. Mat., 30(1-2) (2019), 113–127.
  • [8] S. K. Yadav, S. K. Chaubey, D. L. Suthar, Certain results on almost Kenmotsu (k;m;n)􀀀spaces, Konuralp J. Math., 6(1) (2018) 128–133.
  • [9] H. İ. Yoldas, E. Yasar, A study on N(k)-contact metric manifolds, Balk. J. Geom. Its Appl., 25(1) (2020), 127–140.
  • [10] S. K. Yadav, X. Chen, On h-Einstein N(k)-contact metric manifolds, Bol. Soc. Pran. Mat., 41(3) (2021), 1–13.
  • [11] H. I. Yoldas, Certain Results on N(k)-Contact Metric Manifolds and Torse-Forming Vector Fields, J. Math. Ext., 15 (2021), 1–16.
  • [12] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255—306.
  • [13] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math., Santa Cruz, CA, 1986, 71, Amer. Math. Soc. Providence, RI, (1988), 237–262.
  • [14] T. Chave, G. Valent, Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta, 69 (1996), 344–347.
  • [15] T. Chave, G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys., 478 (1996), 758–778.
  • [16] G. Ayar, M. Yildirim, h-Ricci solitons on nearly Kenmotsu manifolds, Asian-Eur. J. Math., 12(6) (2019), 2040002.
  • [17] C. Calin, M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f -Kenmotsu manifolds, Bull. Malyas. Math. Sci. Soc., 33 (2010), 361–368.
  • [18] Y. J. Suh, S. K. Chaubey, MNI Khan, Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection, Rev. Math. Phys., 36(3) (2024), Paper No. 2450001.
  • [19] S. K. Chaubey, U. C. De, Y. J. Suh, Conformal vector field and gradient Einstein solitons on h-Einstein cosymplectic manifolds, Int. J. Geom. Methods Mod. Phys., 20(8) (2023), Paper No. 2350135, 16 pp.
  • [20] Y. J. Suh, S. K. Chaubey, Ricci solitons on general relativistic spacetimes, Phys. Scr., 98 (2023), 065207.
  • [21] A. Haseeb, S. K. Chaubey, M. A. Khan, Riemannian 3-manifolds and Ricci-Yamabe solitons, Int. J. Geom. Methods Mod. Phys., 20(1) (2023), Paper No. 2350015, 13 pp.
  • [22] S. K. Chaubey, Y. J. Suh, Riemannian concircular structure manifolds, Filomat, 36(19) (2022), 6699–6711.
  • [23] S. K. Chaubey, G. -E. Vˆılcu, Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116(4) (2022), Paper No. 186, 14 pp.
  • [24] A. Haseeb, R. Prasad, Some results on Lorentzian para-Kenmotsu manifolds, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 13(62) (2020), 185–198.
  • [25] A. Haseeb, S. K. Chaubey, Lorentzian para-Sasakian manifolds and ?-Ricci solitons, Kragujev. J. Math., 48(2) (2024), 167–179.
  • [26] H. Öztürk, S. K. Yadav, A note on Ricci and Yamabe solitons on almost Kenmotsu manifolds, Novi Sad J. Math., 53(2) (2023), 223–239.
  • [27] A. Sarkar, G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian-space forms with quasi-Sasakian metric, Africa Maths., 31 (2020), 455–463.
  • [28] H. İ. Yoldaş, A. Haseeb, F. Mofarreh, Certain curvature conditions on Kenmotsu manifolds and ?􀀀h-Ricci solitons, Axioms, 12(2) (2023), 14 pages.
  • [29] C. A. Mantica, L. G. Molonari, Weakly Z -symmetric manifolds, Acta Math. Hunger., 135 (2012), 80–96.
  • [30] M. Ali, A. Haseeb, F. Mofarreh, M. Vasiulla, Z -symmetric manifolds admitting Schouten tensor, Mathematics, 10 (2022), 4293, https://doi.org/10.3390/math10224293.
  • [31] S. K. Chaubey, Trans-Sasakian manifolds satisfying certain conditions, TWMS J. App. Eng. Math., 9(2) (2019), 305–314.
  • [32] S. K. Chaubey, On special weakly Riccisymmetric and generalized Ricci-recurrent trans-Sasakian structures, Thai J. Math., 16(3) (2018), 693–707.
  • [33] A. Barman, I. Unal, Geometry of Kenmotsu manifolds admitting Z-tensor, Bull. Transilv. Univ. Bras., 2(64) (2022), 23–40.
  • [34] U. S. Negi, P. Chauhan, Tensor structures and recurrent Z -forms in Riemannian manifolds, Aryabhatta J. Math. Inf., 14(2) (2022), 153–160.
  • [35] D. G. Prakasha, P. Veeresha, M. Nagaraja, Z -symmetries of e-para-Sasakian 3-manifolds, arXiv:1909.05535v1, (2019).
  • [36] I. Unal, N(k)-contact metric manifolds admitting Z -tensor, KMU J. Eng. Natural Sciences, 2(1) (2020), 64–69.
  • [37] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10, Springer-Verlag, Berlin, Heidelberg, New York, (1987).
  • [38] A. Derdzinski, C. L. Shen, Codazzi tensor fields curvature and Pontryagin forms, Proc. Lond. Math. Soc., 47(1) (1983), 15-–26.
  • [39] F. de Felice, C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, (1990).
  • [40] C. A. Mantica, Y. J. Suh, Pseudo-Q-symmetric Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(5) (2013), 25 pages.
  • [41] K. Yano, Concircular geometry I, Concircular transformation, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.
  • [42] S. K. Yadav, A. Yildiz, Q-curvature tensor on f -Kenmotsu 3-manifolds, Univers. J. Math. Appl., 5(3) (2022), 96-106.
  • [43] M. Yildirim, A new characterization of Kenmotsu manifolds with respect to Q tensor, J. Geom. Phys., 176 (2022), 104498.
  • [44] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes In Math., 509, Springer-Verlag Berlin Heidelberg, (1976).
  • [45] S. K. Yadav, X. Chen, A note on (k;m)-contact metric manifolds, Analele University Oradea Fasc. Matematica, XXIX(2) (2022), 17–28.
  • [46] B. J. Papantoniou, Contact Riemannian manifolds satisfying R(x ;X):R=0 and x 2 (k;m)-nullity distribution, Yokohama Math. J., 40 (1993), 149–161.
  • [47] D. E. Blair, J. S. Kim, M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, Fundam. J. Math. Appl., 3(2) (2020), 94–100.
  • [48] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 40 (1988), 441-448.
  • [49] C. A. Mantica, Y. J. Suh, Pseudo Z- symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys., 9(1) (2012), 1250004.
  • [50] C. A. Mantica, Y. J. Suh, Pseudo Z-symmetric space-times, J. Math. Phys., 55 (2014), 042502.
  • [51] C. A. Mantica, Y. J. Suh, Recurrent Z-forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250059.

Year 2024, , 83 - 92, 23.05.2024

Abhishek Singh , S. K. Chaubey , Sunil Yadav , Shraddha Patel

https://doi.org/10.32323/ujma.1418496

Abstract

References

  • [1] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40 (1988), 441–448.
  • [2] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Isr. J. Math., 91 (1995), 189–214.
  • [3] U. C. De, Y. J. Suh, S. K. Chaubey, Conformal vector fields on almost co-K¨ahler manifolds, Math. Slovaca, 71(6) (2021), 1545–1552.
  • [4] U. C. De, S. K. Chaubey, Y. J. Suh, A note on almost co-K¨ahler manifolds, Int. J. Geom. Methods Mod. Phys., 17(10) (2020), 2050153, 14 pp.
  • [5] S. K. Chaubey, M. A. Khan, A. S. R. Al Kaabi, N(k)-paracontact metric manifolds admitting the Fischer-Marsden conjecture, AIMS Math., 9(1) (2024), 2232–2243.
  • [6] S. K. Chaubey, K. K. Bhaishya, M. D. Siddiqi, Existence of some classes of N(k)-quasi Einstein manifolds, Bol. Soc. Parana. Mat., 39(5) (2021), 145–162.
  • [7] S. K. Chaubey, Certain results on N(k)-quasi Einstein manifolds, Afr. Mat., 30(1-2) (2019), 113–127.
  • [8] S. K. Yadav, S. K. Chaubey, D. L. Suthar, Certain results on almost Kenmotsu (k;m;n)􀀀spaces, Konuralp J. Math., 6(1) (2018) 128–133.
  • [9] H. İ. Yoldas, E. Yasar, A study on N(k)-contact metric manifolds, Balk. J. Geom. Its Appl., 25(1) (2020), 127–140.
  • [10] S. K. Yadav, X. Chen, On h-Einstein N(k)-contact metric manifolds, Bol. Soc. Pran. Mat., 41(3) (2021), 1–13.
  • [11] H. I. Yoldas, Certain Results on N(k)-Contact Metric Manifolds and Torse-Forming Vector Fields, J. Math. Ext., 15 (2021), 1–16.
  • [12] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255—306.
  • [13] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and General Relativity, Contemp. Math., Santa Cruz, CA, 1986, 71, Amer. Math. Soc. Providence, RI, (1988), 237–262.
  • [14] T. Chave, G. Valent, Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta, 69 (1996), 344–347.
  • [15] T. Chave, G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys., 478 (1996), 758–778.
  • [16] G. Ayar, M. Yildirim, h-Ricci solitons on nearly Kenmotsu manifolds, Asian-Eur. J. Math., 12(6) (2019), 2040002.
  • [17] C. Calin, M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f -Kenmotsu manifolds, Bull. Malyas. Math. Sci. Soc., 33 (2010), 361–368.
  • [18] Y. J. Suh, S. K. Chaubey, MNI Khan, Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection, Rev. Math. Phys., 36(3) (2024), Paper No. 2450001.
  • [19] S. K. Chaubey, U. C. De, Y. J. Suh, Conformal vector field and gradient Einstein solitons on h-Einstein cosymplectic manifolds, Int. J. Geom. Methods Mod. Phys., 20(8) (2023), Paper No. 2350135, 16 pp.
  • [20] Y. J. Suh, S. K. Chaubey, Ricci solitons on general relativistic spacetimes, Phys. Scr., 98 (2023), 065207.
  • [21] A. Haseeb, S. K. Chaubey, M. A. Khan, Riemannian 3-manifolds and Ricci-Yamabe solitons, Int. J. Geom. Methods Mod. Phys., 20(1) (2023), Paper No. 2350015, 13 pp.
  • [22] S. K. Chaubey, Y. J. Suh, Riemannian concircular structure manifolds, Filomat, 36(19) (2022), 6699–6711.
  • [23] S. K. Chaubey, G. -E. Vˆılcu, Gradient Ricci solitons and Fischer-Marsden equation on cosymplectic manifolds, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116(4) (2022), Paper No. 186, 14 pp.
  • [24] A. Haseeb, R. Prasad, Some results on Lorentzian para-Kenmotsu manifolds, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 13(62) (2020), 185–198.
  • [25] A. Haseeb, S. K. Chaubey, Lorentzian para-Sasakian manifolds and ?-Ricci solitons, Kragujev. J. Math., 48(2) (2024), 167–179.
  • [26] H. Öztürk, S. K. Yadav, A note on Ricci and Yamabe solitons on almost Kenmotsu manifolds, Novi Sad J. Math., 53(2) (2023), 223–239.
  • [27] A. Sarkar, G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian-space forms with quasi-Sasakian metric, Africa Maths., 31 (2020), 455–463.
  • [28] H. İ. Yoldaş, A. Haseeb, F. Mofarreh, Certain curvature conditions on Kenmotsu manifolds and ?􀀀h-Ricci solitons, Axioms, 12(2) (2023), 14 pages.
  • [29] C. A. Mantica, L. G. Molonari, Weakly Z -symmetric manifolds, Acta Math. Hunger., 135 (2012), 80–96.
  • [30] M. Ali, A. Haseeb, F. Mofarreh, M. Vasiulla, Z -symmetric manifolds admitting Schouten tensor, Mathematics, 10 (2022), 4293, https://doi.org/10.3390/math10224293.
  • [31] S. K. Chaubey, Trans-Sasakian manifolds satisfying certain conditions, TWMS J. App. Eng. Math., 9(2) (2019), 305–314.
  • [32] S. K. Chaubey, On special weakly Riccisymmetric and generalized Ricci-recurrent trans-Sasakian structures, Thai J. Math., 16(3) (2018), 693–707.
  • [33] A. Barman, I. Unal, Geometry of Kenmotsu manifolds admitting Z-tensor, Bull. Transilv. Univ. Bras., 2(64) (2022), 23–40.
  • [34] U. S. Negi, P. Chauhan, Tensor structures and recurrent Z -forms in Riemannian manifolds, Aryabhatta J. Math. Inf., 14(2) (2022), 153–160.
  • [35] D. G. Prakasha, P. Veeresha, M. Nagaraja, Z -symmetries of e-para-Sasakian 3-manifolds, arXiv:1909.05535v1, (2019).
  • [36] I. Unal, N(k)-contact metric manifolds admitting Z -tensor, KMU J. Eng. Natural Sciences, 2(1) (2020), 64–69.
  • [37] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10, Springer-Verlag, Berlin, Heidelberg, New York, (1987).
  • [38] A. Derdzinski, C. L. Shen, Codazzi tensor fields curvature and Pontryagin forms, Proc. Lond. Math. Soc., 47(1) (1983), 15-–26.
  • [39] F. de Felice, C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, Cambridge, (1990).
  • [40] C. A. Mantica, Y. J. Suh, Pseudo-Q-symmetric Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(5) (2013), 25 pages.
  • [41] K. Yano, Concircular geometry I, Concircular transformation, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.
  • [42] S. K. Yadav, A. Yildiz, Q-curvature tensor on f -Kenmotsu 3-manifolds, Univers. J. Math. Appl., 5(3) (2022), 96-106.
  • [43] M. Yildirim, A new characterization of Kenmotsu manifolds with respect to Q tensor, J. Geom. Phys., 176 (2022), 104498.
  • [44] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes In Math., 509, Springer-Verlag Berlin Heidelberg, (1976).
  • [45] S. K. Yadav, X. Chen, A note on (k;m)-contact metric manifolds, Analele University Oradea Fasc. Matematica, XXIX(2) (2022), 17–28.
  • [46] B. J. Papantoniou, Contact Riemannian manifolds satisfying R(x ;X):R=0 and x 2 (k;m)-nullity distribution, Yokohama Math. J., 40 (1993), 149–161.
  • [47] D. E. Blair, J. S. Kim, M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, Fundam. J. Math. Appl., 3(2) (2020), 94–100.
  • [48] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 40 (1988), 441-448.
  • [49] C. A. Mantica, Y. J. Suh, Pseudo Z- symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys., 9(1) (2012), 1250004.
  • [50] C. A. Mantica, Y. J. Suh, Pseudo Z-symmetric space-times, J. Math. Phys., 55 (2014), 042502.
  • [51] C. A. Mantica, Y. J. Suh, Recurrent Z-forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250059.
There are 51 citations in total.

Details

Primary Language English
Subjects Topology
Journal Section Articles
Authors

Abhishek Singh 0009-0007-6784-7395

S. K. Chaubey 0000-0002-3882-4596

Sunil Yadav 0000-0001-6930-3585

Shraddha Patel 0000-0001-9773-9546

Early Pub Date May 11, 2024
Publication Date May 23, 2024
Submission Date January 12, 2024
Acceptance Date April 11, 2024
Published in Issue Year 2024

Cite

APA Singh, A., Chaubey, S. K., Yadav, S., Patel, S. (2024). $\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Universal Journal of Mathematics and Applications, 7(2), 83-92. https://doi.org/10.32323/ujma.1418496
AMA Singh A, Chaubey SK, Yadav S, Patel S. $\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. May 2024;7(2):83-92. doi:10.32323/ujma.1418496
Chicago Singh, Abhishek, S. K. Chaubey, Sunil Yadav, and Shraddha Patel. “$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications 7, no. 2 (May 2024): 83-92. https://doi.org/10.32323/ujma.1418496.
EndNote Singh A, Chaubey SK, Yadav S, Patel S (May 1, 2024) $\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Universal Journal of Mathematics and Applications 7 2 83–92.
IEEE A. Singh, S. K. Chaubey, S. Yadav, and S. Patel, “$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”, Univ. J. Math. Appl., vol. 7, no. 2, pp. 83–92, 2024, doi: 10.32323/ujma.1418496.
ISNAD Singh, Abhishek et al. “$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications 7/2 (May 2024), 83-92. https://doi.org/10.32323/ujma.1418496.
JAMA Singh A, Chaubey SK, Yadav S, Patel S. $\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. 2024;7:83–92.
MLA Singh, Abhishek et al. “$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure”. Universal Journal of Mathematics and Applications, vol. 7, no. 2, 2024, pp. 83-92, doi:10.32323/ujma.1418496.
Vancouver Singh A, Chaubey SK, Yadav S, Patel S. $\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure. Univ. J. Math. Appl. 2024;7(2):83-92.

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