A Kenmotsu metric as a \(\ast\)-conformal Yamabe soliton with torse forming potential vector field. (English) Zbl 1484.53058
Summary: The purpose of the present paper is to deliberate \(\ast\)-conformal Yamabe soliton, whose potential vector field is torse-forming on Kenmotsu manifold. Here, we have shown the nature of the soliton and find the scalar curvature when the manifold admitting \(\ast\)-conformal Yamabe soliton on Kenmotsu manifold. Next, we have evolved the characterization of the vector field when the manifold satisfies \(\ast\)-conformal Yamabe soliton. Also we have embellished some applications of vector field as torse-forming in terms of \(\ast\)-conformal Yamabe soliton on Kenmotsu manifold. We have developed an example of \(\ast\)-conformal Yamabe soliton on 3-dimensional Kenmotsu manifold to prove our findings.
MSC:
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53E99 | Geometric evolution equations |
Keywords:
\(\ast\)-conformal Yamabe soliton; torse forming vector field; conformal Killing vector field; Kenmotsu manifoldReferences:
| [1] | Bagewadi, C. S.; Prasad, V. S., Note on Kenmotsu manifolds, Bull. Cal. Math. Soc., 91, 379-384 (1999) · Zbl 0977.53512 |
| [2] | Barbosa, E.; Ribeiro, E. Jr., On conformal solutions of the Yamabe flow, Arch. Math., 101, 79-89 (2013) · Zbl 1270.53073 · doi:10.1007/s00013-013-0533-0 |
| [3] | Basu, N.; Bhattacharyya, A., Conformal Ricci soliton in Kenmotsu manifold, Global Journal of Advanced Research on Classical and Modern Geometries, 4, 15-21 (2015) |
| [4] | Cao, H. D.; Sun, X.; Zhang, Y., On the structure of gradient Yamabe solitons, Math. Res. Lett., 19, 4, 767-774 (2012) · Zbl 1267.53066 · doi:10.4310/MRL.2012.v19.n4.a3 |
| [5] | Chen, B. Y.: A simple characterization of generalized Robertson-Walker space-times. Gen. Relativity Gravitation, 46, Article ID 1833, 5 pp. (2014) · Zbl 1308.83159 |
| [6] | Chen, B. Y., Classification of torqued vector fields and its applications to Ricci solitons, Kragujevac J. Math., 41, 239-250 (2017) · Zbl 1488.53152 · doi:10.5937/KgJMath1702239C |
| [7] | Cho, J. T.; Kimura, M., Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61, 205-212 (2009) · Zbl 1172.53021 · doi:10.2748/tmj/1245849443 |
| [8] | Ghosh, A., Yamabe soliton and Quasi Yamabe soliton on Kenmotsu manifold, Mathematica Slovaca, 70, 151-160 (2020) · Zbl 1505.53108 · doi:10.1515/ms-2017-0340 |
| [9] | Hamada, T., Real hypersurfaces of complex space forms in terms of Ricci *-tensor, Tokyo J. Math., 25, 473-483 (2002) · Zbl 1046.53037 · doi:10.3836/tjm/1244208866 |
| [10] | Hamilton, R. S., Three Manifold with positive Ricci curvature, J. Differential Geom., 17, 2, 255-306 (1982) · Zbl 0504.53034 · doi:10.4310/jdg/1214436922 |
| [11] | Hamilton, R. S., The Ricci flow on surfaces, Contemporary Mathematics, 71, 237-261 (1988) · Zbl 0663.53031 · doi:10.1090/conm/071/954419 |
| [12] | Kaimakamis, G.; Panagiotidou, K., *-Ricci solitons of real hypersurface in non-flat complex space forms, Journal of Geometry and Physics, 76, 408-413 (2014) · Zbl 1312.53067 · doi:10.1016/j.geomphys.2014.09.004 |
| [13] | Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J., 24, 93-103 (1972) · Zbl 0245.53040 · doi:10.2748/tmj/1178241594 |
| [14] | Schouten, J. A., Ricci Calculus (1954), Berlin: Springer-Verlag, Berlin · Zbl 0057.37803 · doi:10.1007/978-3-662-12927-2 |
| [15] | Singh, A.; Kishor, S., Some types of η-Ricci solitons on Lorentzian para-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 33, 2, 217-230 (2018) · Zbl 1474.53100 |
| [16] | Roy, S.; Bhattacharyya, A., Conformal Ricci solitons on 3-dimensional trans-Sasakian manifold, Jordan Journal of Mathematics and Statistics, 13, 1, 89-109 (2020) · Zbl 1474.53310 |
| [17] | Roy, S., Dey, S., Bhattacharyya, A., et al.: *-conformal η-Ricci soliton on Sasakian manifold, accepted for publication in Asian-European Journal of Mathematics, doi:10.1142/S1793557122500358 (2021) |
| [18] | Roy, S.; Dey, S.; Bhattacharyya, A., Yamabe Solitons on (LCS)_n-manifolds, Journal of Dynamical Systems & Geometric Theories, 18, 2, 261-279 (2020) · Zbl 1499.53164 · doi:10.1080/1726037X.2020.1868100 |
| [19] | Roy, S., Dey, S., Bhattacharyya, A.: Some results on η-Yamabe solitons in 3-dimensional trans-Sasakian manifold. arXiv:2001.09271v2 [math.DG] (2020) |
| [20] | Roy, S.; Dey, S.; Bhattacharyya, A., A Kenmotsu metric as a conformal η-Einstein soliton, Carpathian Mathematical Publications, 13, 1, 110-118 (2021) · Zbl 1473.53056 · doi:10.15330/cmp.13.1.110-118 |
| [21] | Roy, S., Dey, S., Bhattacharyya, A.: Conformal Yamabe soliton and *-Yamabe soliton with torse forming potential vector field, accepted for publication in Matematički Vesnik (2020) |
| [22] | Tachibana, S., On almost-analytic vectors in almost Kählerian manifolds, Tohoku Math. J., 11, 247-265 (1959) · Zbl 0090.38603 |
| [23] | Tanno, S., The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J., 21, 21-38 (1969) · Zbl 0188.26705 |
| [24] | Topping, P., Lecture on the Ricci Flow (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1105.58013 · doi:10.1017/CBO9780511721465 |
| [25] | Venkatesha; Naik, D. M.; Kumara, H. A., *-Ricci solitons and gradient almost *-Ricci solitons on Kenmotsu manifolds, Math. Slovava, 69, 6, 1447-1458 (2019) · Zbl 1505.53063 · doi:10.1515/ms-2017-0321 |
| [26] | Yano, K., Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo, 16, 195-200 (1940) · Zbl 0024.08102 |
| [27] | Yano, K., On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20, 340-345 (1944) · Zbl 0060.39102 |
| [28] | Yano, K.; Chen, B. Y., On the concurrent vector fields of immersed manifolds, Kodai Math. Sem. Rep., 23, 343-350 (1971) · Zbl 0221.53049 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
