Browse > Article
http://dx.doi.org/10.5831/HMJ.2018.40.2.325

RICCI SOLITONS ON RICCI PSEUDOSYMMETRIC (LCS)n-MANIFOLDS  

Hui, Shyamal Kumar (Department of Mathematics, The University of Burdwan)
Lemence, Richard S. (Institute of Mathematics, College of Science, University of Philippines)
Chakraborty, Debabrata (Department of Mathematics, Sidho Kanho Birsha University)
Publication Information
Honam Mathematical Journal / v.40, no.2, 2018 , pp. 325-346 More about this Journal
Abstract
The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci pseudosymmetric, $W_3$-Ricci pseudosymmetric, conharmonic Ricci pseudosymmetric, conformal Ricci pseudosymmetric $(LCS)_n$-manifolds to be shrinking, steady and expanding. We also construct an example of concircular Ricci pseudosymmetric $(LCS)_3$-manifold whose metric is Ricci soliton.
Keywords
Ricci soliton; (LCS)_n$-manifold; Ricci pseudosymmetric manifold; concircular curvature tensor; projective curvature tensor; $W_3$-curvature tensor; conharmonic curvature tensor; conformal curvature tensor;
Citations & Related Records
Times Cited By KSCI : 4  (Citation Analysis)
연도 인용수 순위
1 R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306.   DOI
2 R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity, Contemp. Math., 71 (1988), 237-262.
3 S. K. Hui, On $\phi$-pseudo symmetries of $(LCS)_n$-manifolds, Kyungpook Math. J., 53 (2013), 285-294.   DOI
4 S. K. Hui and M. Atceken, Contact warped product semi-slant submanifolds of $(LCS)_n$-manifolds, Acta Univ. Sapientiae Mathematica, 3(2) (2011), 212-224.
5 S. K. Hui and D. Chakraborty, Some types of Ricci solitons on $(LCS)_n$-manifolds, J. Math. Sci. Advances and Applications, 37 (2016), 1-17.
6 S. K. Hui and D. Chakraborty, Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field, New Trends in Math. Sciences, 4 (2016), 263-269.   DOI
7 C. S. Bagewadi and G. Ingalahalli, Ricci solitons in Lorentzian ${\alpha}$-Sasakian manifolds, Acta Math. Acad. Paeda. Nyire., 28 (2012), 59-68.
8 R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44(1) (1992), 1-34.
9 S. R. Ashoka, C. S. Bagewadi and G. Ingalahalli, A geometry on Ricci solitons in $(LCS)_n$-manifolds, Diff. Geom.-Dynamical Systems, 16 (2014), 50-62.
10 S. R. Ashoka, C. S. Bagewadi and G. Ingalahalli, Certain results on Ricci solitons in ${\alpha}$-Sasakian manifolds, Hindawi Publ. Corporation, Geometry, Vol. 2013, Article ID 573925, 4 pages.
11 M. Atceken, On geometry of submanifolds of $(LCS)_n$-manifolds, Int. J. Math. and Math. Sci., 2012, doi:10.1155/2012/304647.   DOI
12 M. Ateceken and S. K. Hui, Slant and pseudo-slant submanifolds of LCS-manifolds, Czechoslovak Math. J., 63 (2013), 177-190.   DOI
13 A. A. Shaikh and T. Q. Binh, On weakly symmetric $(LCS)_n$-manifolds, J. Adv. Math. Studies, 2 (2009), 75-90.
14 A. A. Shaikh and S. K. Hui, On some classes of generalized quasi-Einstein man-ifolds, Commun. Korean Math. Soc., 24(3) (2009), 415-424.   DOI
15 A. A. Shaikh and S. K. Hui, On generalized ${\phi}$-recurrent $(LCS)_n$-manifolds, AIP Conf. Proc., 1309 (2010), 419-429.
16 A. A. Shaikh, Y. Matsuyama and S. K. Hui, On invariant submanifold of $(LCS)_n$-manifolds, J. of Egyptian Math. Soc., 24 (2016), 263-269.   DOI
17 R. Sharma, Certain results on k-contact and (k, ${\mu}$)-contact manifolds, J. of Geom., 89 (2008), 138-147.   DOI
18 M. M. Tripathi, Ricci solitons in contact metric manifolds, arxiv:0801.4221[Math.DG] (2008).
19 K. Yano, Concircular geometry I, concircular transformations, Proc. Imp. Acad. Tokyo, 16 (1940), 195-200.   DOI
20 H. G. Nagaraja and C. R. Premalatha, Ricci solitons in Kenmotsu manifolds, J. Math. Analysis, 3 (2012), 18-24.
21 G. Perelman, Ricci flow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, 2003, 1-22.
22 D. Narain and S. Yadav, On weak concircular symmetries of $(LCS)_{2n+1}$-manifolds, Global J. Sci. Frontier Research, 12 (2012), 85-94.
23 B. O'Neill, Semi Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
24 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math/0211159, 2002, 1-39.
25 G. P. Pokhariyal, Curvature tensors and their relativistic significance III, Yokohama Math. J., 21 (1973), 115-119.
26 D. G. Prakasha, On Ricci ${\eta}$-recurrent $(LCS)_n$-manifolds, Acta Univ. Apulensis, 24 (2010), 109-118.
27 A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Math. J., 43 (2003), 305-314.
28 A. A. Shaikh, Some results on $(LCS)_n$-manifolds, J. Korean Math. Soc., 46 (2009), 449-461.   DOI
29 A. A. Shaikh and H. Ahmad, Some transformations on $(LCS)_n$-manifolds, Tsukuba J. Math., 38 (2014), 1-24.   DOI
30 A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat., 1 (2005), 129-132.   DOI
31 A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes II, American J. Appl. Sci., 3(4) (2006), 1790-1794.   DOI
32 S. K. Hui and D. Chakraborty, Para-Sasakian manifolds and Ricci solitons, Ilirias J. of Math., 6 (2017), 25-34.
33 A. A. Shaikh, T. Basu and S. Eyasmin, On locally ${\phi}$-symmetric $(LCS)_n$-manifolds, Int. J. of Pure and Appl. Math., 41(8) (2007), 1161-1170.
34 A. A. Shaikh, T. Basu and S. Eyasmin, On the existence of ${\phi}$-recurrent $(LCS)_n$-manifolds, Extracta Mathematicae, 23(1) (2008), 71-83.
35 I. Mihai and R. Rosca, On Lorentzian para-Sasakian manifolds, Classical Anal., World Sci. Publ., Singapore, (1992), 155-169.
36 S. K. Hui and D. Chakraborty, ${\eta}$-Ricci solitons on ${\eta}$-Einstein $(LCS)_n$-manifolds, Acta Univ. Palac. Olom., Fac. Rer. Nat., Math., 55(2) (2016), 101-109.
37 S. K. Hui and D. Chakraborty, Infinitesimal CL-transformations on Kenmotsu manifolds, Bangmod Int. J. Math. and Comp. Sci., 3 (2017), 1-9.
38 S. K. Hui and D. Chakraborty, Ricci almost solitons on Concircular Ricci pseudosymmetric ${\beta}$-Kenmotsu manifolds to appear in Hacettepe J. of Math. and Stat.
39 S. K. Hui and R. S. Lemence, Ricci pseudosymmetric generalized quasi-Einstein manifolds, Sut J. Math., 51 (2015), 67-85.
40 S. K. Hui, R. S. Lemence and D. Chakraborty, Ricci solitons on three dimensional generalized Sasakian-space-forms, Tensor Society, N. S., 76 (2015), 75-83.
41 S. K. Hui, R. Prasad and D. Chakraborty, Ricci solitons on Kenmotsu Manifolds with respect to quarter symmetric non-metric ${\phi}$-connection, Ganita, Bharata Ganita Parishad, 67 (2017), 195-204.
42 Y. Ishii, On conharmonic transformations, Tensor N. S., 11 (1957), 73-80.
43 S. K. Hui, S. S. Shukla and D. Chakraborty, ${\eta}$-Ricci solitons on ${\eta}$-Einstein Kenmotsu manifolds, Global J. Adv. Res. Clas. Mod. Geom., 6(1) (2017), 1-6.
44 S. K. Hui, S. Uddin and D. Chakraborty, Infinitesimal CL-transformations on $(LCS)_n$-manifolds, Palestine J. Math., 6 (Special Issue: II) 6 (Special Issue: II) (2017), 190-195.
45 K. Matsumoto, On Lorentzian almost paracontact manifolds, Bull. of Yamagata Univ. Nat. Sci., 12 (1989), 151-156.
46 S. K. Hui, S. Uddin and D. Chakraborty, Generalized Sasakian-space-forms whose metric is ${\eta}$-Ricci almost solitons, Diff. Geom. and Dynamical Systems, 19 (2017), 45-55.
47 G. Ingalahalli and C. S. Bagewadi, Ricci solitons in ${\alpha}$-Sasakian manifolds, Hindawi Publishing Corporation, ISRN Geometry, Vol. 2012, Article ID 421384, 13 pages.
48 B. Jahanara, S. Haesen, Z. Senturk and L. Verstraelen, On the parallel transport of the Ricci curvatures, J. Geom. Phys., 57 (2007), 1771-1777.   DOI
49 W. Kuhnel, Conformal transformations between Einstein spaces, conformal geometry, 105-146, Aspects Math., E12, Vieweg, Braun-schweig, 1988.
50 C. L. Bejan and M. Crasmareanu, Ricci solitons in manifolds with quasi constant curvature, Publ. Math. Debrecen, 78 (2011), 235-243.   DOI
51 A. M. Blaga, ${\eta}$-Ricci solitons on para-kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 1-13.
52 S. Deshmukh, H. Al-Sodais and H. Alodan, A note on Ricci solitons, Balkan J. Geom. Appl.,16 (2011), 48-55.
53 S. Chandra, S. K. Hui and A. A. Shaikh, Second order parallel tensors and Ricci solitons on $(LCS)_n$-manifolds, Commun. Korean Math. Soc., 30 (2015), 123-130.   DOI
54 B. Y. Chen and S. Deshmukh, Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), 13-21.
55 U. C. De and A. A. Shaikh, Differential Geometry of Manifolds, Narosa Publishing House Pvt. Ltd., New Delhi, 2007.
56 R. Deszcz, On Ricci-pseudosymmetric warped products, Demonstratio Math., 22 (1989), 1053-1065.