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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.19 no.2 Temuco jun. 2017 


Some geometric properties of η Ricci solitons and gradient Ricci solitons on (lcs) n -manifolds

S. K. Yadav1 

S. K. Chaubey2 

D. L. Suthar3 

1 Poornima college of Engineering, Department of Mathematics, ISI-6, RIICO, Institutional Area, Sitapura, Jaipur-302022, Rajasthan, India. e-mail:

2 Shinas College of Technology, Section of Mathematics, Department of Information Technology, Shinas, P.O. Box 77 Postal Code 324, Oman. e-mail:,

3 Wollo University, Department of Mathematics, P. O. Box: 1145, Dessie, South Wollo, Amhara Region, Ethiopia.


In the context of para-contact Hausdorff geometry η-Ricci solitons and gradient Ricci solitons are considered on manifolds. We establish that on an (LCS)n-manifold (M, ϕ, ξ, η, 𝑔), the existence of an ηRicci soliton implies that (M, 𝑔) is quasi-Einstein. We find conditions for Ricci solitons on an (LCS)n-manifold (M, ϕ, ξ, η, 𝑔) to be shrinking, steady and expanding. At the end we show examples of such manifolds with η-Ricci solitons.

Keywords and Phrases: η-Ricci solitons; gradient Ricci solitons; (LCS)n-manifold.


En el contexto de geometría para-contacto Hausdorff, consideramos η-Ricci solitones y Ricci solitones gradientes en variedades. Establecemos que en una (LCS)n-variedad (M, ϕ, ξ, η, 𝑔), la existencia de un η-Ricci solitón implica que (M, 𝑔) es casi-Einstein. Encontramos condiciones para que los Ricci solitones en una (LCS)n-variedad (M, ϕ, ξ, η, 𝑔) sean contractivos, estables o expansivos. Al concluir, mostramos ejemplos de dichas variedades con η-Ricci solitones.

2010 AMS Mathematics Subject Classification: 53C25, 53C15, 53C21.


[1] M. Atceken, On geometry of submanifold of (LCS)n-manifolds, Int. J. Math. Sci., (2012), Art. ID304647. [ Links ]

[2] M. Atceken and S. K. Hui, Slant and pseudo-slant submanifold of (LCS)n-manifolds, Czechoslovak, Math. J., 63(1), (2013), 177-190. [ Links ]

[3] C. S. Bagewadi and G. Ingalahalli, Ricci soliton in Lorentzian α-Sasakian manifolds, Acta Math. Academiae Paedagogical Nyiregyhaziensis 28(1), (2012), 59-68. [ Links ]

[4] C. S. Bagewadi, G. Ingalahalli and S. R. Ashoka, A Study of Ricci soliton in Kenmotsu manifolds, ISRN Geometry, (2013), Article ID 412593. [ Links ]

[5] C. L. Bejan and M. Crasmareanu, Second order parallel tensor and Ricci solitons in 3-dimensional normal para-contact geometry, Anal. Global Anal. Geom. DOI:10.1007/s10455-014-9414-4. [ Links ]

[6] J. T. Cho and M. Kimura, Ricci soliton and real hypersurfaes in a complex space form, Tohoku Math. J., 61(2),(2009), 205-212. [ Links ]

[7] S. K. Chaubey, On weakly m-projectively symmetric manifolds, Novi Sad J. Math., 42 (1) (2012), 67-79. [ Links ]

[8] S. K. Chaubey and R. H. Ojha, On the m-projective curvature tensor of a Kenmotsu manifold, Differential Geometry-Dynamical systems, 12 (2010), 52-60. [ Links ]

[9] S. K. Chaubey, Some properties of LP-Sasakian manifolds equipped with m-projective curvature tensor, Bulletin of Mathematical Analysis and apllications, 3 (4) (2011), 50-58. [ Links ]

[10] S. K. Chaubey and C. S. Prasad, On generalized ϕ-recurrent Kenmotsu manifolds, TWMS J. App. Eng. Math., 5(1) (2015), 1-9. [ Links ]

[11] S. K. Chaubey, S. Prakash and R. Nivas, Some properties of m-projective curvature tensor in Kenmotsu manifolds, Bulletin of Math. Analysis and Applications, 4 (3) (2012), 48-56. [ Links ]

[12] B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, AMS, Providence, RI, USA, (2006). [ Links ]

[13] G. Calvaruso and D. Perrone, Geometry of H-paracontcat metric manifolds, (2013), arXiv; 1307.7662v1. [ Links ]

[14] C. Calin and M. Crasmareanu, Eta-Ricci soliton on Hopfhypersurfaces in complex space forms, Revnue Roumaine de Mathematiquespures et applications, 57(1), (2012), 55-63. [ Links ]

[15] C. Calim and M. Crasmareanu, Form the Eisenhart problem to Ricci soliton in f-Kenmotsu manifolds, Bull. Malaysian Math. Sci. Soc., 33(3), (2010), 361-368. [ Links ]

[16] B. Y. Chen and S. Deshmikh, Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19(1), (2014), 13-21. [ Links ]

[17] S. Chandra, S. K. Hui and A. A. Shaikh, Second order parallel Tensors and Ricci solitons on (LCS)-manifolds, Korean Math. Soc., 30(2), (2015), 123-130. [ Links ]

[18] O. Chodosh and F. T. H. Fong, Rotational symmetry of conical Kähler Ricci solitons, (2013), arXiv: 1304.0277v2. [ Links ]

[19] A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasakian manifolds and toric Sasakian-Einstein manifolds, J. Diff. Geom., 83(3), (2009), 585-636. [ Links ]

[20] C. He and M. Zhu, The Ricci soliton on Sasakian manifold, (2011), arXiv:1109.4404v2. [ Links ]

[21] S. K. Hui and M. Atceken, Contact warped product semi-slant submanifolds of (LCS)n-manifolds, Acta Univ. Saoientiae Math., 3(2), (2011), 212-224. [ Links ]

[22] S. K. Hui, On ϕ-pseudo symmetries of (LCS)-manifolds, Kyungpook Math.J., 53(2), (2013), 285-294. [ Links ]

[23] S. K. Chaubey, Existence of N(k)-quasi Einstein manifolds, Facta Universitatis (NIS) Ser. Math. Inform., 32 (3), (2017), 369-385. [ Links ]

[24] R. S. Hamilton, The Ricci flow on surfaces, Math. and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, AMS (1988). [ Links ]

[25] G. Ingalahalli and C. S. Bagewadi, Ricci soliton on α-Sasakian manifolds, ISRN Geometry, (2012), Article ID 421384, pages 13. [ Links ]

[26] I. Mihai, and R. Rosca, On Lorentzian para-Sasakian manifolds, Classical Anal., (1992),155-169. [ Links ]

[27] K. Matsumoto, On Lorentzain almost paracontact manifolds, Bull. Yamagata Univ. Nature. Sci., 12, (1989), 151-156. [ Links ]

[28] H. G. Nagaraja and C. R. Premalatha, Ricci soliton in Kenmotsu manifolds, J. Math. Anal., 3(2), (2012), 18-24. [ Links ]

[29] B. O'Neill, Semi Riemannian geometry with applications to relativity, Academic Press, New York, (1983). [ Links ]

[30] D. Narain and S. Yadav, On weak Symmetric of Lorentzian concircular structure manifolds, CUBO A Mathematical Journal, 15(2), (2013), 33-42. [ Links ]

[31] R. Sharma, Certain results on K-contact and (κ ,μ)-contact manifolds, J. of Geometry, 89, (2008), 138-147. [ Links ]

[32] A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat. 1(2), (2005), 129-132. [ Links ]

[33] A. A. Shaikh and K. K. Baishya, On concircular spacetimes II, Amer. J. Appl. Sci., 3(4), (2006), 1790-1794. [ Links ]

[34] A. A. Shaikh, On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Math. J., 43, (2003), 305-314. [ Links ]

[35] A. A. Shaikh, T. Basu and S. Eyasmin, On locally ϕ-symmetric (LCS)n-manifolds, Int. J. Pure Appl. Math. 41(8), (2007), 1161-1170. [ Links ]

[36] A. Shaikh, T. Basu and S. Eyasmin, On the existence of ϕ-recurrent (LCS)n-manifolds, Extr. Math., 23(1), (2008), 71-83. [ Links ]

[37] T. Takahashi, Sasakian ϕ-symmetric spaces, Tohoku Math. J., 29(1), (1977), 91-113. [ Links ]

[38] M. M. Tripathi, Ricci solitons in contact metric manifolds, [ Links ]

[39] S. Yadav, D. L. Suthar and P. K. Dwivedi, Some results on (LCS)2n+1-manifolds, IAMURE, International journal of Mathematics, Engineering & Technology, 6, (2013), 73-84. [ Links ]

[40] S. Yadav and P. K. Dwivedi, On (LCS)n-manifolds satisfying certain conditions on D-conformal curvature tensor, Global Journal of Frontier science Research, Mathematics Decision Science, 14, (2012). [ Links ]

[41] S. Yadav, P. K. Dwivedi and D. L. Suthar, On (LCS)2n+1-manifolds satisfying certain conditions on the concircular curvature tensor, Thai Journal of Mathematics, 9, (2011), 597-603. [ Links ]

[42] S. Yadav, D. L. Suthar and Mebrahtu Hailu, On extended generalized φ-recurrent (LCS)2n+1-manifolds, Bol. Soc. Paran. Mat. (3s.) v. 37 (2), (2019), 9-21. [ Links ]

[43] S. K. Yadav, S. K. Chaubey and D. L. Suthar, Certain geometric properties of η-Ricci soliton on η-Einstein para-Kenmotsu manifolds, Palestine Journal of Mathematics, 7 (2), 2018, ??. [ Links ]

[44] K. Yano, Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo, 16, (1940), 195-200. [ Links ]

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