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http://dx.doi.org/10.4134/BKMS.b181175

RICCI SOLITONS AND RICCI ALMOST SOLITONS ON PARA-KENMOTSU MANIFOLD  

Patra, Dhriti Sundar (Department of Mathematics Birla Institute of Technology Mesra)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.5, 2019 , pp. 1315-1325 More about this Journal
Abstract
The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.
Keywords
Ricci soliton; Ricci almost soliton; Einstein manifold; paracontact metric manifold; para-Kenmotsu manifold;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 J. T. Cho and R. Sharma, Contact geometry and Ricci solitons, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 6, 951-960. https://doi.org/10.1142/S0219887810004646   DOI
2 O. Chodosh and F. T.-H. Fong, Rotational symmetry of conical Kahler-Ricci solitons, Math. Ann. 364 (2016), no. 3-4, 777-792. https://doi.org/10.1007/s00208-015-1240-x   DOI
3 K. Erken and C. Murathan, A study of three-dimensional paracontact (${\tilde{k}},\;{\tilde{\mu}},\;{\tilde{\nu}}$)-spaces, Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 7, 1750106, 35 pp. https://doi.org/10.1142/S0219887817501067   DOI
4 A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons Fractals 44 (2011), no. 8, 647-650. https://doi.org/10.1016/j.chaos.2011.05.015   DOI
5 A. Ghosh, An $\eta$-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen 82 (2013), no. 3-4, 591-598. https://doi.org/10.5486/PMD.2013.5344   DOI
6 A. Ghosh and D. S. Patra, The k-almost Ricci solitons and contact geometry, J. Korean Math. Soc. 55 (2018), no. 1, 161-174. https://doi.org/10.4134/JKMS.j170103   DOI
7 G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math. 55 (2011), no. 2, 697-718 (2012). http://projecteuclid.org/euclid.ijm/1359762409   DOI
8 A. Ghosh and D. S. Patra, ∗-Ricci soliton within the frame-work of Sasakian and ($k,\;{\mu}$)-contact manifold, Int. J. Geom. Methods Mod. Phys. 15 (2018), no. 7, 1850120, 21 pp. https://doi.org/10.1142/S0219887818501207   DOI
9 A. Ghosh and R. Sharma, Sasakian metric as a Ricci soliton and related results, J. Geom. Phys. 75 (2014), 1-6. https://doi.org/10.1016/j.geomphys.2013.08.016   DOI
10 S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173-187. https://doi.org/10.1017/S0027763000021565   DOI
11 V. Martin-Molina, Local classification and examples of an important class of paracontact metric manifolds, Filomat 29 (2015), no. 3, 507-515. https://doi.org/10.2298/FIL1503507M   DOI
12 D. S. Patra, Ricci solitons and paracontact geometry, accepted in Mediterr. J. Math. (2019).
13 A. Perrone, Some results on almost paracontact metric manifolds, Mediterr. J. Math. 13 (2016), no. 5, 3311-3326. https://doi.org/10.1007/s00009-016-0687-7   DOI
14 S. Pigola, M. Rigoli, M. Rimoldi, and A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 757-799.
15 R. Sharma, Certain results on K-contact and ($k,\;{\mu}$)-contact manifolds, J. Geom. 89 (2008), no. 1-2, 138-147. https://doi.org/10.1007/s00022-008-2004-5   DOI
16 Y. Wang, Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds, J. Korean Math. Soc. 53 (2016), no. 5, 1101-1114. https://doi.org/10.4134/JKMS.j150416   DOI
17 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
18 Y. Wang and X. Liu, Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese J. Math. 19 (2015), no. 1, 91-100. https://doi.org/10.11650/tjm.19.2015.4094   DOI
19 K. Yano, Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.
20 S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom. 36 (2009), no. 1, 37-60. https://doi.org/10.1007/s10455-008-9147-3   DOI
21 B. Cappelletti-Montano, A. Carriazo, and V. Martin-Molina, Sasaki-Einstein and paraSasaki-Einstein metrics from ($k,\;{\mu}$)-structures, J. Geom. Phys. 73 (2013), 20-36. https://doi.org/10.1016/j.geomphys.2013.05.001   DOI
22 S. Zamkovoy and V. Tzanov, Non-existence of flat paracontact metric structures in dimension greater than or equal to five, Annuaire Univ. Sofia Fac. Math. Inform. 100 (2011), 27-34.
23 C. L. Bejan and M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann. Global Anal. Geom. 46 (2014), no. 2, 117-127. https://doi.org/10.1007/s10455-014-9414-4   DOI
24 A. M. Blaga, Eta-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), no. 1, 1-13.
25 G. Calvaruso and A. Perrone, Ricci solitons in three-dimensional paracontact geometry, J. Geom. Phys. 98 (2015), 1-12. https://doi.org/10.1016/j.geomphys.2015.07.021   DOI
26 G. Calvaruso and D. Perrone, Geometry of H-paracontact metric manifolds, Publ. Math. Debrecen 86 (2015), no. 3-4, 325-346.   DOI