CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS
Abstract
We prove that if an $\eta$-Einstein para-Kenmotsu manifold admits a $\eta$-Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a $\eta$-Ricci soliton is Einstein if its potential vector field $V$ is infinitesimal paracontact transformation or collinear with the Reeb vector field. Further, we prove that if a para-Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits $\eta$-Ricci soliton and satisfy our results. We also have studied $\eta$-Ricci soliton in 3-dimensional normal almost paracontact metric manifolds and we show that if in a 3-dimensional normal almost paracontact metric manifold with $\alpha, \beta $ = constant, the metric is $\eta$-Ricci soliton, where potential vector field $V$ is collinear with the characteristic vector field $\xi$, then the manifold is $\eta$-Einstein manifold.
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Barros, A. and Ribeiro, Jr. E.: Some characterizations for compact almost Ricci solitons,
Proc. Amer. Math. Soc., 140(3), 1033-1040 (2012).
Barros, A., Batista, R. and Ribeiro Jr, E.: Compact almost Ricci solitons with constant
scalar curvature are gradient, Monatsh Math., 174(1), 29-39 (2014).
Cho, J.T. and Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form,
Tohoku Mathematical Journal, Second Series, 61(2), 205-212 (2009).
Blaga, A. M.: Almost η-Ricci solitons in (LCS)n-manifolds, Bull. Belgian Math. Soc. -
Simon Stevin, 25(5), 641-653 (2018).
Blaga, A.M.: η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl., 20(1),
-13 (2015).
Blaga, A.M. and Ozgur, C.: Almost η-Ricci and almost η-Yamabe solitons with torseforming
potential vector field. arXiv preprint arXiv:2003.12574, 2020.
Calin, C. and Crasmareanu, M.: η-Ricci solitons on Hopf hypersurfaces in complex space
forms, Rev. Roumaine Math. Pures Appl., 57(1), 55-63 (2012).
Calvaruso G. and Perrone, D.: Geometry of H-paracontact metric manifolds, Publ. Math.
Debrecen, 86, 325-346 (2015).
Calvaruso, G. and Perrone, A.: Ricci solitons in three-dimensional paracontact geometry,
J. Geom. Phys. 98, 1-12 (2015).
Cho, J. T. and Sharma, R.: Contact geometry and Ricci solitons, Int. J. Geom. Methods
Math. Phys., 7, 951-960 (2010).
Dey, S. and Roy, S.: ∗-η-Ricci Soliton within the framework of Sasakian manifold, Journal
of Dynamical Systems & Geometric Theories, Vol-18(2), pp-163-181 (2020).
Dacko, P., On almost paracosymplectic manifolds, Tsukuba J. Math. 28, no.1, 193-213
(2004).
Erdem, S., On almost (para)contact (hyperbolic) metric manifolds and harmonicity of
(’; ’0) holomorphic maps between them, Houston J. Math., 28, 21-45 (2002).
Ganguly, D., Dey, S., Ali, A., and Bhattacharyya, A.: Conformal Ricci soliton and QuasiYamabe soliton on generalized Sasakian space form, Journal of Geometry and Physics, Vol.
(2021) 104339, 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).
Ghosh, A.: An η-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen,
(3-4), 591-598 (2013).
Hamilton, R.S.: The Ricci ow on surfaces, volume 71. Amer. Math. Soc., 1988.
Kenmotsu, K.: A class of almost contact Riemannian manifolds, T^ohoku Math. J. 24,
-103 (1972).
Kaneyuki, S., Willams, F. L.: Almost paracontact and parahodge structure on manifolds,
Nagoya Math. J., 99, 173-187, 1985.
Naik, D.M. and Venkatesha, V.: η-Ricci soliton and almost η-Ricci soliton on para-Sasakian
manifolds, Int. J. Geom. Methods Mod. Phys., 16(9), 1950134 (2019).
Patra, D.S.: Ricci soliton and paracontact geometry, Mediterr. J. Math., 16:137 (2019).
Patra, D.S.: Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold, Bull.
Korean Math. Soc., 56(5) , 1315-1325 (2019).
Perrone, A.: Some results on almost paracontact metric manifolds, Mediterr. J. Math.,
(5), 3311-3326 (2016).
Prktas, S.Y and Keles, S.: Ricci solitons in 3-dimensional normal almost paracontact metric
manifolds, International Electronic Journal of Geometry.,Volume 8 No. 2 pp. 34-45 (2015).
Pigola, S., Rigoli, M., Rimoldi, M. and Setti, A.: Ricci almost solitons, Ann. Scuola. Norm.
Sup. Pisa. CL Sc., (5) Vol. X , 757-799 (2011).
Roy, S., Dey, S. and Bhattacharyya, A.: Conformal Einstein soliton within the framework
of para-K¨ahler manifold, arXiv:2005.05616v1 [math.DG], Differential Geometry-Dynamical
Systems, Vol.23, pp. 235-243 (2021).
Roy, S., Dey, S. and Bhattacharyya, A.: Yamabe Solitons on (LCS)n-manifolds, Journal of
Dynamical Systems & Geometric Theories, Vol-18(2), pp-261-279 (2020).
Roy, S., Dey, S. and Bhattacharyya, A.: Some results on η-Yamabe Solitons in 3-
dimensional trans-Sasakian manifold, accepted for publication in Carpathian Mathematical
Publications(2021), arXiv:2001.09271v2 [math.DG].
Roy, S., Dey, S. and Bhattacharyya, A.: A Kenmotsu metric as a conformal η-Einstein soliton, Carpathian Mathematical Publications, 13(1), 110-118,
https://doi.org/10.15330/cmp.13.1.110-118 (2021).
Roy, S., Dey, S. and Bhattacharyya, A.: Conformal Yamabe soliton and ∗-Yamabe soliton
with torse forming potential vector field, Matematiˇ cki Vesnik, Vol: 73(4) (2021), pp:282-292.
Roy, S., Dey, S. Bhattacharyya, A. and Hui, S. K.: ∗-Conformal η-Ricci soliton on
Sasakian manifold, Asian-European Journal of Mathematics, Vol. 15, No. 2(2022) 2250035,
https://doi.org/10.1142/S1793557122500358, 2250035(2021).
Sarkar, S., Dey, S. and Chen, X.: Certain results of conformal and ∗-conformal Ricci soliton on para-cosymplectic and para-Kenmotsu manifolds, Filomat 35:15 (2021), 5001{5015,
https://doi.org/10.2298/FIL2115001S.
Sarkar, S., Dey, S. and Bhattacharyya, A: Ricci solitons and certain related metrics on
-dimensional trans-Sasakian manifold, arXiv:2106.10722v1 [math.DG](2021).
Sharma, R.: Certain results on K-contact and (κ; µ)-contact manifolds, J. Geom., 89(1-2),
-147 (2008).
Wang, Y. and Liu, X.: Ricci solitons on three-dimensional η-Einstein almost Kenmotsu
manifolds, Taiwanese J. Math., 19(1), 91-100 (2015).
Welyczko, J.: Legendre curves in 3-dimensional Normal almost paracontact metric manifolds, Result. Mth. 54 (2009), 377-387.
We lyczko, J., Slant curves in 3-dimensional normal almost paracontact metric manifolds,
Mediterr. J. Math., 11(3), 965-978 (2014).
Yano, K.: Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.
Yano, K. and Kon, M.: Structures on Manifolds, Series in Pure Mathematics 3, World
Scientific Pub. Co., Singapore, 1984.
Zamkovoy, S.: Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom.,
(1), 37-60 (2009).
Zamkovoy, S.: On para-Kenmotsu Manifolds, Filomat 32(14), (2018).
DOI: https://doi.org/10.22190/FUMI220210025D
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