Characterization of Almost \(\eta\) -Ricci Solitons With Respect to Schouten-van Kampen Connection on Sasakian Manifolds

Tugba Mert *

Department of Mathematics, University of Sivas Cumhuriyet, 58140, Sivas, Turkey.

Mehmet Atceken

Department of Mathematics, University of Aksaray, 68100, Aksaray, Turkey.

Pakize Uygun

Department of Mathematics, University of Aksaray, 68100, Aksaray, Turkey.

*Author to whom correspondence should be addressed.


Abstract

In this paper, we investigate Sasakian manifolds that admit almost \(\eta\) -Ricci solitons with respect to the Schouten-van Kampen connection using certain curvature tensors. Concepts of Ricci pseudosymmetry for Sasakian manifolds admitting \(\eta\)-Ricci solitons are introduced based on the selection of specific curvature tensors such as Riemann, concircular, projective, pseudo-projective, M-projective, and W2 tensors. Subsequently, necessary conditions are established for a Sasakian manifold admitting \(\eta\)-Ricci soliton with respect to the Schouten-van Kampen connection to be Ricci semisymmetric, based on the choice of curvature tensors. Characterizations are then derived, and classifications are made under certain conditions.

Keywords: Ricci-pseudosymmetric manifold, \(\eta\)-Ricci soliton, Schouten-van Kampen connection


How to Cite

Mert, T., Atceken, M., & Uygun, P. (2024). Characterization of Almost \(\eta\) -Ricci Solitons With Respect to Schouten-van Kampen Connection on Sasakian Manifolds. Asian Journal of Mathematics and Computer Research, 31(1), 64–75. https://doi.org/10.56557/ajomcor/2024/v31i18585

Downloads

Download data is not yet available.

References

Perelman G. The entropy formula for the Ricci ow and its geometric applications. 2002;1-39. DOI: http://arXiv.org/abs/math/0211159

Perelman G. Ricci ow with surgery on three manifolds. 2003;1-22. Available: http://arXiv.org/abs/math/0303109

Sharma R. Certain results on k-contact and (k; )-contact manifolds. J. Geom. 2008;89:138-147.

Ashoka SR, Bagewadi CS, Ingalahalli G. Certain results on Ricci solitons in (alpha) sasakian manifolds. Hindawi Publ. Corporation, Geometry; 2013. Article ID 573925

Ashoka SR, Bagewadi CS, Ingalahalli G. A geometry on Ricci solitons in (LCS)n manifolds. Di . Geom.- Dynamical Systems. 2014;16:50-62.

Bagewadi CS, Ingalahalli G. Ricci solitons in Lorentzian-Sasakian manifolds. Acta Math. Acad. Paeda. Nyire. 2012;28:59-68.

Ingalahalli G, Bagewadi CS. Ricci solitons in (alpha)Sasakian manifolds. ISRN Geometry; 2012. Article ID 421384

Bejan CL, Crasmareanu M. Ricci solitons in manifolds with quasi-contact curvature. Publ. Math. Debrecen. 2011;78:235-243.

Blaga AM. (eta) - Ricci solitons on para-Kenmotsu manifolds. Balkan J. Geom. Appl. 2015;20:1-13.

Chandra S, Hui, SK, Shaikh AA. Second order parallel tensors and Ricci solitons on (LCS)n-manifolds. Commun. Korean Math. Soc. 2015;30:123-130.

Chen BY, Deshmukh S. Geometry of compact shrinking Ricci solitons. Balkan J. Geom. Appl. 2014;19:13- 21.

Deshmukh S, Al-Sodais H, Alodan H. A note on Ricci solitons. Balkan J. Geom. Appl. 2011;16:48-55.

He C, Zhu M. Ricci solitons on Sasakian manifolds; 2011. arxiv:1109.4407V2, [Math DG]

Atceken M, Mert T, Uygun P. Ricci-Pseudosymmetric (LCS)n -manifolds admitting almost (eta)

Ricci solitons. Asian Journal of Math. and Computer Research. 2022:29(2);23-32.

Nagaraja H, Premalatta CR. Ricci solitons in Kenmotsu manifolds. J. Math. Analysis, 2012:3(2);18-24.

Tripathi MM. Ricci solitons in contact metric manifolds; 2008. arxiv:0801,4221 V1, [Math DG]

Schouten JA, Van Kampen ER. Zur Einbettungs-und Krummungstheorie nichtholonomer Gebilde, Math. Ann. 1939:103;752-783.

Vranceanu G. Sur Quelques points De La Theorie Des Espaces Non Holonomes, Bull. Fac. St.Cernauti. 1931:5;177-205.

Bejancu A. Schouten-van Kampen. Vranceanu connections on foliated manifolds. Anale Stinti ce Ale Universitati. AL. I. CUZA" IASI, Tomul LII, Mathematica. 2006:37-60.

Olszak Z. The Schouten Van-Kampen Ane connection adapted to an almost(para) contact metric

structure. Publications De'linstitut Mathematique. 2013:94(108);31-42.

Sasaki S. On di erentiable manifolds with certain structures which are closely related to almos contact structure. Tohoku Math J. 1960:12;456-476.

Cartan E. Sur une classe remarquable D'espaces de Riemannian, Bull. Soc. Math. France. 1926:54;214 264.

Takahasi T. Sasakian '-Symmetryc Spaces. Tohoku Math J. 1977:29;91-113.

Ghosh G. On Schouten-van Kampen connection in Sasakian manifolds. Bol. Soc. Paran. Mat. 2018:36(4);171-182.

Blair DE. Contact manifolds in Riemannian geometry. Lecture Notes in Math., Springer Verlag. 1967;509.

Boyer, Charles P, Galickib K. Sasakian geomatry, oxford mathematical monographs. Oxford University Press, Oxford, MR 2382957, Zbl. 2008;1155.53002.

Tachibana S. On harmonic tensors in compact Sasakian spaces. Tohoku Math J. 1965:17(2);271-284.

Futaki A, Ono H, Wang G. Transverse Kahler geometry of Sasakian manifolds and Toric Sasaki einstein manifolds. J. Di . Geom, 2009:83;585-636.

Martelli D, Sparks J, Yau ST. Sasaki einstein manifolds and volume minimization. Commun. Math. Phys., 2007:280;611-673.

Cho JT, Kimura M. Ricci solitons and real Hypersurfaces in a complex space form. Tohoku Math. J. 2009:61(2);205-212.