On the Almost $\eta-$Ricci Solitons on Pseudosymmetric Lorentz Generalized Sasakian Space Forms

Year 2023, , 43 - 52, 01.07.2023

Tuğba Mert , Mehmet Atçeken

https://doi.org/10.32323/ujma.1236596

Abstract

In this paper, we consider Lorentz generalized Sasakian space forms admitting almost $\eta-$Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of \ Lorentz generalized Sasakian space forms admitting $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, $\mathcal{M-}$projective, $W_{1}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz generalized Sasakian space form admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.

Keywords

Ricci-pseudosymmetric Manifold, η-Ricci Soliton, Generalized Lorentz Sasakian Space Form

References

• [1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Mathematics, (2002), 1–39.
• [2] G. Perelman, Ricci flow with surgery on three manifolds, http://arXiv.org/abs/math/0303109, (2003), 1–22.
• [3] R. Sharma, Certain results on k-contact and (k;m)􀀀contact manifolds, J. Geom., 89 (2008), 138–147.
• [4] S. R. Ashoka, C. S. Bagewadi, G. Ingalahalli, Certain results on Ricci Solitons in a􀀀Sasakian manifolds, Hindawi Publ. Corporation, Geometry, 2013, Article ID 573925, (2013), 4 Pages.
• [5] S. R. Ashoka, C. S. Bagewadi, G. Ingalahalli, A geometry on Ricci solitons in (LCS)n manifolds, Diff. Geom.-Dynamical Systems, 16 (2014), 50–62.
• [6] C. S. Bagewadi, G. Ingalahalli, Ricci solitons in Lorentzian-Sasakian manifolds, Acta Math. Acad. Paeda. Nyire., 28 (2012), 59–68.
• [7] G. Ingalahalli, C. S. Bagewadi, Ricci solitons in a􀀀Sasakian manifolds, ISRN Geometry, 2012, Article ID 421384, (2012), 13 Pages.
• [8] C. L. Bejan, M. Crasmareanu, Ricci Solitons in manifolds with quasi-contact curvature, Publ. Math. Debrecen, 78 (2011), 235–243.
• [9] A. M. Blaga, h􀀀Ricci solitons on para-kenmotsu manifolds, Balkan J. Geom. Appl., 20 (2015), 1–13.
• [10] S. Chandra, S. K. Hui, A. A. Shaikh, Second order parallel tensors and Ricci solitons on (LCS)n-manifolds, Commun. Korean Math. Soc., 30 (2015), 123–130.
• [11] B. Y. Chen, S. Deshmukh, Geometry of compact shrinking Ricci solitons, Balkan J. Geom. Appl., 19 (2014), 13–21.
• [12] S. Deshmukh, H. Al-Sodais, H. Alodan, A note on Ricci solitons, Balkan J. Geom. Appl., 16 (2011), 48–55.
• [13] C. He, M. Zhu, Ricci solitons on Sasakian manifolds, arxiv:1109.4407V2, [Math DG], (2011).
• [14] M. Atc¸eken, T. Mert, P. Uygun, Ricci-Pseudosymmetric (LCS)n􀀀manifolds admitting almost h􀀀Ricci solitons, Asian Journal of Math. and Computer Research, 29(2), (2022), 23–32.
• [15] H. Nagaraja, C. R. Premalatta, Ricci solitons in Kenmotsu manifolds, J. Math. Analysis, 3(2), (2012), 18–24.
• [16] M. M. Tripathi, Ricci solitons in contact metric manifolds, arxiv:0801,4221 V1, [Math DG], (2008).
• [17] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Volume 203 of Progress in Mathematics, Birkhauser Boston, Inc., Boston, MA, USA, 2nd edition, 2010.
• [18] P. Alegre, D.E. Blair, A. Carriazo, Generalized Sasakian space form, Israel Journal of Mathematics, 141 (2004), 157–183.
• [19] P. Alegre, A. Carriazo, Semi-Riemannian generalized Sasakian space forms, Bulletin of the Malaysian Mathematical Sciences Society, 41(1) (2001), 1–14.
• [20] J. T Cho, M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61 (2) (2009), 205–212.