Abstract
In the present paper, we initiate the study of \(*\)-\(\eta \)-Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a \(*\)-\(\eta \)-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost \(*\)-\(\eta \)-Ricci soliton. Next, we deliberate \(*\)-\(\eta \)-Ricci soliton admitting \((\kappa ,\mu )^\prime \)-almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to \({\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n\). Finally we present some examples to decorate the existence of \(*\)-\(\eta \)-Ricci soliton, gradient almost \(*\)-\(\eta \)-Ricci soliton on Kenmotsu manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhäuser, Boston (2010)
Cǎlin, C., Crasmareanu, M.: From the Eisenhart problem to Ricci solitons in \(f\)-Kenmotsu manifolds. Bull. Malays. Math. Sci. Soc. (2) 33(3), 361–368 (2010)
Chen, X.: Real hypersurfaces with \(*\)-Ricci solitons of non-flat complex space forms. Tokyo J. Math. 41, 433-3e451 (2018)
Cho, J.T., Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form. Töhoku Math. J. 61, 205–212 (2009)
Dai, X.: Non-existence of \(*\)-Ricci solitons on \((\kappa ,\mu )\)-almost cosymplectic manifolds. J. Geom. 110(2), 30 (2019)
Dai, X., Zhao, Y., De, U.C.: \(*\)-Ricci soliton on \((\kappa ,\mu )^\prime \)-almost Kenmotsu manifolds. Open Math. 17, 874–882 (2019)
Dey, S., Roy, S.: \(*\)-\(\eta \)-Ricci Soliton within the framework of Sasakian manifold. J. Dyn. Syst. Geom. Theor. 18(2), 163–181 (2020)
Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93, 46–61 (2009)
Duggal, K.L.: Almost Ricci solitons and physical applications. Int. Electron. J. Geom. 10(2), 1–10 (2017)
Ganguly, D., Dey, S., Ali, A., Bhattacharyya, A.: Conformal Ricci soliton and Quasi-Yamabe soliton on generalized Sasakian space form. J. Geom. Phys. (2021). https://doi.org/10.1016/j.geomphys.2021.104339
Ghosh, A.: Kenmotsu 3-metric as a Ricci soliton. Chaos Solitons Fractals 44, 647–650 (2011)
Ghosh, A.: Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold. Carpathian Math. Publ. 11(1), 59–69 (2019)
Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)
Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math J. 4(1), 1–27 (1981)
Kaimakanois, G., Panagiotidou, K.: \(*\)-Ricci solitons of real hypersurface in non-flat comlex space forms. J. Geom. Phys. 86, 408–413 (2014)
Kenmotsu, K.: A class of almost contact Riemannian manifolds. Töhoku Math. J. 24, 93–103 (1972)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint arXiv:math.DG/0211159
Roy, S., Dey, S., Bhattacharyya, A., Hui, S.K.: \(*\)-Conformal \(\eta \)-Ricci Soliton on Sasakian manifold. Asian-Eur. J. Math. 10, 10 (2021). https://doi.org/10.1142/S1793557122500358
Roy, S., Dey, S., Bhattacharyya, A.: Conformal Einstein soliton within the framework of para-Kähler manifold. In: Differential Geometry-Dynamical Systems, vol. 23, pp. 235–243 (2021). arXiv:2005.05616v1 [math.DG]
Roy, S., Dey, S., Bhattacharyya, A.: A Kenmotsu metric as a conformal \(\eta \)-Einstein soliton. Carpathian Math. Publ. 13(1), 110–118 (2021). https://doi.org/10.15330/cmp.13.1.110-118
Roy, S., Dey, S., Bhattacharyya, A.: Conformal Yamabe soliton and \(*\)-Yamabe soliton with torse forming potential vector field. Matematički Vesnik (2021). arXiv:2105.13885v1 [math.DG]
Sarkar, S., Dey, S., Bhattacharyya, A.: Ricci solitons and certain related metrics on 3-dimensional trans-Sasakian manifold. arXiv:2106.10722v1 [math.DG] (2021)
Sarkar, S., Dey, S., Chen, X.: Certain results of conformal and \(*\)-conformal Ricci soliton on para-cosymplectic and para-Kenmotsu manifolds. Filomat (2021)
Sharma, R.: Certain results on \(K\)-contact and \((\kappa,\mu )\)-contact manifolds. J. Geom. 89, 138–147 (2008)
Tachibana, S.: On almost-analytic vectors in almost Kaehlerian manifolds. Tohoku Math. J. 11, 247–265 (1959)
Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Töhoku Math. J. 21, 21–38 (1969)
Venkatesha, V., Naik, D.M., Kumara, H.A.: \(*\)-Ricci solitons and gradient almost \(*\)-Ricci solitons on Kenmotsu manifolds. Math. Slovaca 69(6), 1447–1458 (2019)
Wang, Y.: Contact 3-manifolds and \(*\)-Ricci soliton. Kodai Math. J. 43(2), 256–267 (2020)
Wang, W.: Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds. J. Korean Math. Soc. 53, 1101–1114 (2016)
Woolgar, E.: Some applications of Ricci flow in physics. Can. J. Phys. 86(4), 645–651 (2008)
Yano, K.: Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1. Marcel Dekker Inc, New York (1970)
Acknowledgements
The second author was financially supported by UGC Senior Research Fellowship of India, Sr. No. 2061540940. Ref. No:21/06/2015(i)EU-V.
Author information
Authors and Affiliations
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dey, S., Sarkar, S. & Bhattacharyya, A. \(*\)-\(\eta \)-Ricci soliton and contact geometry. Ricerche mat (2021). https://doi.org/10.1007/s11587-021-00667-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-021-00667-0
Keywords
- Ricci flow
- \(\eta \)-Ricci soliton
- \(*\)-\(\eta \)-Ricci
- Gradient almost \(*\)-\(\eta \)-Ricci soliton
- Kenmotsu manifold
- \((\kappa , \mu )^\prime \)-almost Kenmotsu manifold