[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/CKMS.c190121

∗-RICCI SOLITONS AND ∗-GRADIENT RICCI SOLITONS ON 3-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS

Dey, Dibakar (Department of Pure Mathematics University of Calcutta)
Majhi, Pradip (Department of Pure Mathematics University of Calcutta)

Publication Information

Communications of the Korean Mathematical Society / v.35, no.2, 2020 , pp. 625-637 More about this Journal

Abstract

The object of the present paper is to characterize 3-dimensional trans-Sasakian manifolds of type (α, β) admitting ∗-Ricci solitons and ∗-gradient Ricci solitons. Under certain restrictions on the smooth functions α and β, we have proved that a trans-Sasakian 3-manifold of type (α, β) admitting a ∗-Ricci soliton reduces to a β-Kenmotsu manifold and admitting a ∗-gradient Ricci soliton is either flat or ∗-Einstein or it becomes a β-Kenmotsu manifold. Also an illustrative example is presented to verify our results.

Keywords

Trans-Sasakian manifolds; ${\ast}$ -Ricci soliton; ${\ast}$ -gradient Ricci soliton; ${\ast}$ -Einstein manifold;

Citations & Related Records

Reference

1	D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
2	D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J. (2) 29 (1977), no. 3, 319-324. https://doi.org/10.2748/tmj/1178240602 DOI
3	D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat. 34 (1990), no. 1, 199-207. https://doi.org/10.5565/PUBLMAT_34190_15 DOI
4	U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J. 43 (2003), no. 2, 247-255.
5	S. Deshmukh, Trans-Sasakian manifolds homothetic to Sasakian manifolds, Mediterr. J. Math. 13 (2016), no. 5, 2951-2958. https://doi.org/10.1007/s00009-015-0666-4 DOI
6	S. Deshmukh and M. M. Tripathi, A note on trans-Sasakian manifolds, Math. Slovaca 63 (2013), no. 6, 1361-1370. https://doi.org/10.2478/s12175-013-0176-4 DOI
7	T. Dutta, N. Basu, and A. Bhattacharyya, Almost conformal Ricci solituons on 3-dimensional trans-Sasakian manifold, Hacet. J. Math. Stat. 45 (2016), no. 5, 1379-1391.
8	A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35-58. https://doi.org/10.1007/BF01796539 DOI
9	A. Ghosh and D. S. Patra, *-Ricci soliton within the frame-work of Sasakian and ( ${\kappa},{\mu}$ )-contact manifold, Int. J. Geom. Methods Mod. Phys. 15 (2018), no. 7, 1850120, 21 pp. https://doi.org/10.1142/S0219887818501207 DOI
10	T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci *-tensor, Tokyo J. Math. 25 (2002), no. 2, 473-483. https://doi.org/10.3836/tjm/1244208866 DOI
11	R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. http://projecteuclid.org/euclid.jdg/1214436922 DOI
12	G. Kaimakamis and K. Panagiotidou, *-Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86 (2014), 408-413. https://doi.org/10.1016/j.geomphys.2014.09.004 DOI
13	J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985), no. 3-4, 187-193.
14	K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. (2) 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594 DOI
15	J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl. (4) 162 (1992), 77-86. https://doi.org/10.1007/BF01760000 DOI
16	M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. https://doi.org/10.2969/jmsj/01430333 DOI
17	S. Tachibana, On almost-analytic vectors in almost-Kahlerian manifolds, Tohoku Math. J. (2) 11 (1959), 247-265. https://doi.org/10.2748/tmj/1178244584 DOI
18	M. Turan, U. C. De, and A. Yildiz, Ricci solitons and gradient Ricci solitons in threedimensional trans-Sasakian manifolds, Filomat 26 (2012), no. 2, 363-370. https://doi.org/10.2298/FIL1202363T DOI
19	L. Vanhecke and D. Janssens, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27. http://projecteuclid.org/euclid.kmj/1138036310 DOI
20	Y. Wang, Ricci solitons on almost Kenmotsu 3-manifolds, Open Math. 15 (2017), no. 1, 1236-1243. https://doi.org/10.1515/math-2017-0103 DOI
21	Y. Wang and X. Liu, Ricci solitons on three-dimensional ${\eta}$ -Einstein almost Kenmotsu manifolds, Taiwanese J. Math. 19 (2015), no. 1, 91-100. https://doi.org/10.11650/tjm.19.2015.4094 DOI
22	K. Yano, Integral formulas in Riemannian geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.