Remarks on Some Soliton Types with Certain Vector Fields

Year 2022, , 146 - 159, 28.07.2022

Halil İbrahim Yoldaş

https://doi.org/10.54974/fcmathsci.1077820

Cited By: 3

Abstract

This paper mainly aims to investigate some soliton kinds with certain vector fields on Riemannian manifolds and gives some notable geometric results as regards such vector fields. Also, in this paper some special tensors that have an important place in Riemannian geometry are discussed and given some significant links between these tensors. Finally, an example that supports one of our results is given.

Keywords

Ricci soliton, Yamabe soliton, Conformal quadratic Killing tensor, $\mathcal{Z}$-curvature tensor.

References

• Barbosa E., Ribeiro E., On conformal solutions of the Yamabe flow, Archiv der Mathematik, 101, 79-89, 2013.
• Blaga A.M., Özgür C., Almost η -Ricci and almost η -Yamabe solitons with torse-forming vector field, Quaestiones Mathematicae, 45(1), 143-163, 2022.
• Chen B.-Y., Classification of Torqued vector fields and its applications to Ricci solitons, Kragujevac Journal of Mathematics, 41(2), 39-250, 2017.
• Chen B.-Y., Some results on concircular vector fields and their applications to Ricci solitons, Bulletin of the Korean Mathematical Society, 52(5), 1535-1547, 2015.
• Chen B.-Y., Deshmukh S., Yamabe and quasi-Yamabe solitons on Euclidean submanifolds, Mediterranean Journal of Mathematics, 15(5), Article: 194, 2018.
• Cho J.T., Kimura M., Ricci solitons and real hypersurfaces in a complex space form, Tohoku Mathematical Journal, 61, 205-212, 2009.
• Crasmareanu M., Parallel tensors and Ricci solitons in N(k)−quasi Einstein manifolds, Indian Journal of Pure and Applied Mathematics, 43, 359-369, 2012.
• Duggal K.L., Affine conformal vector fields in semi-Riemannian manifolds, Acta Applicandae Mathematicae, 23, 275-294, 1991.
• Duggal K.L., Almost Ricci solitons and physical applications, International Electronic Journal of Geometry, 10(2), 1-10, 2017.
• Duggal K.L., Symmetry inheritance in Riemannian manifolds with applications, Acta Applicandae Mathematicae, 31, 225-247, 1993.
• Hamilton R.S., The Ricci flow on surfaces, Contemporary Mathematics, 71, 237-262, 1988.
• Hamilton R.S., Three-manifolds with positive Ricci curvature, Journal of Differential Geometry, 17, 255 306, 1982.
• Katzin G.H., Levine J., Davis W.R., Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, Journal of Mathematical Physics, 10(4), 617-628, 1969.
• Majhi P., Ghosh G., Concircular vectors field in (k,μ)-contact metric manifolds, International Electronic Journal of Geometry, 11(1), 52-56, 2018.
• Mantica C.A., Molinari L.G., Weakly Z -symmetric manifolds, Acta Mathematica Hungarica, 135(1-2), 80-96, 2012.
• Meriç Ş.E., Kılıç E., Riemannian submersions whose total manifolds admit a Ricci soliton, International Journal of Geometric Methods in Modern Physics, 16(12), 1950196, 2019.
• Naik D.M., Venkatesha V., η -Ricci solitons and almost η -Ricci solitons on Para-Sasakian manifolds, International Journal of Geometric Methods in Modern Physics, 16(9), 1950134, 2019.
• Patra D.S., Ricci solitons and paracontact geometry, Mediterranean Journal of Mathematics, 16(6), Article:137, 2019.
• Pigola S., Rigoli M., Rimoldi M., Setti A., Ricci almost solitons, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, (10)4, 757-799, 2011.
• Romero A., Sanchez M., Projective vector fields on Lorentzian manifolds, Geometriae Dedicata, 93, 95-105, 2002.
• Sharma R., On certain results on K-contact and (k,μ)-contact manifolds, Journal of Geometry, 89(1-2), 138-147, 2008.
• Wald R.M., General Relativity, University of Chicago Press, 1984.
• Walker M., Penrose R., On quadratic first integrals of the geodesic equations for type {22} spacetimes, Communications in Mathematical Physics, 18, 265-274, 1970.
• Yano K., Integral Formulas in Riemannian Geometry, Marcel Dekker, 1970.
• Yano K., Kon M., On torse-forming direction in a Riemannian space, Proceedings of the Imperial Academy, 20, 340-345, 1944.
• Yoldaş H.İ., Meriç Ş.E., Yaşar E., On generic submanifold of Sasakian manifold with concurrent vector field, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1983-1994, 2019.
• Yoldaş H.İ., Meriç Ş.E., Yaşar E., On submanifolds of Kenmotsu manifold with Torqued vector field, Hacettepe Journal of Mathematics and Statistics, 49(2), 843-853, 2020.
• Zengin F.Ö., On Riemannian manifolds admitting W2 -curvature tensor, Miskolc Mathematical Notes, 12(2), 289-296, 2011.