Korean J. Math. Vol. 32 No. 2 (2024) pp.315-328
DOI: https://doi.org/10.11568/kjm.2024.32.2.315

A $(k,\mu)$-contact metric manifold as an $\eta-$Einstein soliton

Main Article Content

Arup Kumar Mallick
Arindam Bhattacharyya

Abstract

 The aim of the paper is to study an $\eta$-Einstein soliton on $(2n+1)$-dimensional $(k,\mu)$-contact metric manifold. At first, we establish various results related to $(2n+1)$-dimensional $(k,\mu)$-contact metric manifold that exhibit an $\eta$-Einstein soliton. Next we study some curvature conditions admitting an $\eta$-Einstein soliton on $(2n+1)$-dimensional $(k,\mu)$-contact metric manifold. Furthermore, we consider specific conditions associated with an $\eta$-Einstein soliton on $(2n+1)$-dimensional $(k,\mu)$-contact metric manifold. Finally, we show the existance of an $\eta$-Einstein soliton on $(k,\mu)$-contact metric manifold.



Article Details

References

[1] N. Basu and A. Bhattacharyya, Conformal Ricci soliton in Kenmotsu manifold, Global Journal of Advanced Research on classical and Modern Geometrices, 4 (1) (2015), 15–21. Google Scholar

[2] A. M. Blaga, On gradient η−Einstein solitons, Kragujevac Journal of Mathematics 42 (2) (2018), 229–237. https://doi.org/10.5937/KGJMATH1802229B Google Scholar

[3] D. E. Blair, J. S. Kim and M. M. Tripathi, On the concircular curvature tensor of a contact metric manifold, J.Korean Math.Soc. 42 (2005), 883–992. https://doi.org/10.4134/JKMS.2005.42.5.883 Google Scholar

[4] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), 189–214. Google Scholar

[5] D .E. Blair, Contact manifolds in Riemannian geometry, Lecture notes in Mathematics, Springer-Verlag, Berlin, Vol.509 (1976). Google Scholar

[6] D. E. Blair, Two remarks on contact metric structures, Tohoku Math.J. 29 (1977), 319–324. https://doi.org/10.2748/TMJ/1178240602 Google Scholar

[7] E. Boeckx, P. Bueken and L. Vanhecke, φ-symmetric contact metric spaces, Glasgow Math. J. 41 (1999), 409–416. Google Scholar

[8] E. Boecks, A full classification of contact metric (k, μ)-spaces, Illinois J. Math. 44 (2000), 212– 219. Google Scholar

[9] C. Calin and M. Crasmareanu, η−Ricci solitons on Hopf hypersurfaces in a complex space forms, Revue Roumaine de Math.Pures et Appl. 57 (1) (2012), 53–63. Google Scholar

[10] G. Catino and I. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 66–94. https://doi.org/10.1016/j.na.2015.10.021 Google Scholar

[11] J.T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J 61 (2) (2009), 205–212. https://doi.org/10.2748/tmj/1245849443 Google Scholar

[12] Mohd. Danish Siddiqi, Conformal η−Ricci solitons in δ−Lorentzian trans Sasakian manifolds, International Journal of Maps in Mathematics 1 (1) (2018), 15–34. Google Scholar

[13] U. C. De and A. A. Shaikh, Complex Manifolds and Contact Manifolds, 142–143. Google Scholar

[14] A. E. Fischer, An introduction to conformal Ricci flow, class. Quantum. Grav. 21 (2004), 171– 218. https://doi.org/10.1088/0264-9381/21/3/011 Google Scholar

[15] R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry (Cambridge, MA, 1993) 2 7-136, International Press, Cambridge, MA, 1995. Google Scholar

[16] R. S. Hamilton, Three manifolds with positive Ricci curvature, J.Differential Geom. (1982), Vol.17, isu.2, 255–306. https://doi.org/10.1086/wp.17.4.1180866 Google Scholar

[17] D. Kar, P. Majhi and U. C. De, η-Ricci solitons on 3-dimensional N(k)-contact metric manifolds, Acta Univ. Apulensis 54 (2018), 71–88. Google Scholar

[18] T. Koufogiorgos, Contact metric manifolds, Annals of Global Analysis and Geometry 11 (1993), 25–34. https://doi.org/10.1007/BF00773361 Google Scholar

[19] T. Koufogiorgos, On a class of contact Riemannian 3-manifolds, Results in Mathematics 27 (1995), 51–62. https://doi.org/10.1007/BF03322269 Google Scholar

[20] E. M. Patterson, Some theorems on Ricci recurrent spaces, J. London Math. Soc. 27 (1952), 287–295. Google Scholar

[21] S. Tanno, The topology of contact Riemannian manifolds, Illinois J. Math. 12 (1968), 700–717. https://doi.org/10.1215/IJM/1256053971 Google Scholar

[22] M. M. Tripathi and J. S. Kim, On the concircular curvature tensor of a (k, μ)-manifold, Balkan J Geo Appl 2 (2004), 104–114. Google Scholar

[23] C. Udriste, On contact 3-structures, Bull. Univ. Brasov 16 (1974), 85–92. https://doi.org/10.1007/s002090100279 Google Scholar

[24] K. Yano, Concircular geometry I. Concircular transformations, Proc. Impe. Acad., Tokyo., 16 (1940), 195–200. https://doi.org/10.3792/PIA/1195579139 Google Scholar

[25] K. Yano, On torse-forming directions in Riemannian spaces, Proc. Impe. Acad., Tokyo., 20 (1944), 701–705. https://doi.org/10.3792/PIA/1195572958 Google Scholar