Some Inequalities for Ricci Solitons

Abstract

We deal with a submanifold of a Ricci soliton $(\bar{M},\bar{g},V,\lambda)$ and obtain

that under what conditions such a submanifold is Ricci soliton. Moreover, we establish some inequalities for Ricci solitons to obtain the relationships between the intrinsic or extrinsic invariants.

Keywords

• Ricci soliton,Riemannian manifold,Curvature

References

1. Barros, A, Vieira Gomes, JN, Ribeiro, E. \emph{Immersion of almost Ricci solitons into a Riemannian manifold }, Math. Cont. \textbf{40}(2011), 91--102. Bejan, C.L., Crasmareanu, M., \emph{Ricci solitons in manifolds with quasi-constant curvature}, Publ. Math. Debrecen. \textbf{78}(2011), 235--243. Bejan, C.L., Crasmareanu, M., \emph{Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry}, Anal. Glob. Anal. Geom. \textbf{46}(2014), 117--128. Blaga, A.M., Perkta\c{s}, S.Y., \emph{Remarks on almost $\eta-$Ricci solitons in ($\varepsilon$)-para Sasakian manifolds}, (2018), arXiv:1804.05389v1. Blaga, A.M., Perkta\c{s}, S.Y, Acet, B.E., Erdo\u{g}an, F.E., \emph{$\eta-$Ricci solitons in ($\varepsilon$)-almost paracontact metric manifolds}, (2017), arXiv: 1707.07528v2. Calin, C., Crasmareanu, M., \emph{From the Eisenhart problem to Ricci solitons in $f$-Kenmotsu manifolds}, Bull. Malays. Math. Sci. Soc. \textbf{33}(2010), 361--368. Besse, A.L., Einstein manifolds, Berlin-Heidelberg-New York: Spinger-Verlag, 1987. Chen, B.Y., \emph{Concircular vector fields and pseudo-K\"{a}hler manifolds}, Kragujevac J. Math. \textbf{40}(2016), 7--14. Chen B.Y., Deshmukh, S., \emph{Ricci solitons and concurrent vector Field}, Balkan J. Geom. Its Appl. \textbf{20}(2015), 14--25. Chen, B.Y., \emph{Ricci solitons on Riemannian submanifolds}. In: Mihai A, Mihai I, editors. RIGA-Proceedings of the Conference; 19-21 May; Bucharest, Romania. University of Bucharest Press, (2014) 30--45. Chen, B.Y., Deshmukh, S., \emph{Classification of Ricci solitons on Euclidean hypersurfaces}, Int. J. Math. \textbf{ 25}(2014), 22 pp. Hamilton, R.S., \emph{The Ricci flow on surfaces, Mathematics and General Relativity(Santa Cruz, CA, 1986)}, Contemp. Math. Amer. Math. Soc. \textbf{71}(1988), 237--262. Perelman, G., \emph{The Entropy formula for the Ricci flow and its geometric applications}, (2002) arXiv math/0211159. Tripathi, M.M., \emph{Certain basic inequalities for submanifolds in $(\kappa,\mu)$ -space}, Recent advances in Riemannian and Lorentzian geometries, Baltimore: MD, 2003. Deshmukh, S., Alodan, H., Al-Sodais, H., \emph{A note on Ricci solitons}, Balkan J. Geom. Its Appl. 16(2011) 48--55. Perkta\c{s}, S.Y., Kele\c{s}, S., \emph{Ricci solitons in 3-dimensional normal almost paracontact metric manifolds}, Int. Elect. J. Geom. 8(2015), 34--45.