On generic submanifold of Sasakian manifold with concurrent vector field

Yıl 2019, Cilt: 68 Sayı: 2, 1983 - 1994, 01.08.2019

Halil İbrahim Yoldaş Şemsi Eken Meriç Erol Yaşar

https://doi.org/10.31801/cfsuasmas.445788

Cited By: 6

Öz

In the present paper, we deal with the generic submanifold admitting a Ricci soliton in Sasakian manifold endowed with concurrent vector field. Here, we find that there exists never any concurrent vector field on the invariant distribution D of generic submanifold M. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D^{⊥} of M are Einstein. Finally, we give a characterization for a generic submanifold of Sasakian manifold to be a gradient Ricci soliton.

Anahtar Kelimeler

Sasakian manifold, Ricci soliton, concurrent vector field

Kaynakça

• P. Alegre, Semi-Invariant Submanifolds of Lorentzian Sasakian Manifolds, \emph{Demonstratio Math.} 44:2(2011), 33-38.
• M. Atçeken, S. Uddin, Semi-invariant submanifolds of a normal almost paracontact manifold, \emph{Filomat} 31(15) (2017), 4875-4887.
• C. S. Bagewadi, G. Ingalahalli, Ricci solitons in Lorentzian $\alpha-$Sasakian manifolds, \emph{Acta Math. Acad. Paedagog. Nyh\'{a}zi. (N.S)} 28(1) (2012), 59-68.
• C. L. Bejan, M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-simensional normal paracontact geometry, \emph{Ann. Glob. Anal. Geom.} 46(2014), 117-127.
• A. Bejancu, N. Papaghiuc, Semi-invariant submanifolds of Sasakian manifold, \emph{An. St. Univ. AI. I. Cuza. Iasi} 27(1981), 163-170.
• A. Bejancu, N. Papaghiuc, Semi-invariant submanifolds of a Sasakian space form, \emph{Colloq. Math.} 48(1984), 77-88.
• A. Bejancu, Geometry of CR-submanifolds, Mathematics and Its Applications (East European Series), (1986).
• D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 509(1976).
• B.-Y. Chen, S. Deshmukh, Ricci solitons and concurrent vector fields, \emph{Balkan J. Geom. Appl.} 20(1) (2015), 14-25.
• B.-Y. Chen, Classification of torqued vector fields and its applications to Ricci solitons, \emph{Kragujevac J. Math.} 41(2) (2017), 239-250.
• K. L. Duggal, D. L. Jin, Generic lightlike submanifolds of an indefinite SasakianmManifold, \emph{Int. Electron. J. Geom.} 5(1) (2012), 108-119.
• Ghosh, Certain contact metrics as Ricci almost solitons, \emph{Results Maths.} 65(2014), 81-94.
• A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, \emph{Chaos, Solitons \& Fractals} 44(8) (2011), 647-650.
• R. S. Hamilton, Three-manifolds with positive Ricci curvature, \emph{J. Diff. Geom.} 17(2) (1982), 255-306.
• C. He, M. Zhu, The Ricci solitons on Sasakian manifolds, \emph{arxiv:1109.4407v2.} 2011.
• G. Ingalahalli, C. S. Bagewadi, Ricci solitons in $\alpha-$Sasakian manifolds, \emph{ISRN Geometry} vol.2012, Article ID 421384, 13 Pages, (2012).
• M. A. Khan, M. Z. Khan, Totally umbilical semi-invariant submanifolds of a nearly cosymplectic manifold, \emph{Filomat} 20(2) (2006), 33-38.
• M. E. A. Mekki, A. M. Cherif, Generalised Ricci solitons on Sasakian manifolds, \emph{Kyungpook Math. J.} 57(4) (2017), 677-682.
• A. Mihai and R. Roşça, On a class of Einstein space-time manifolds, \emph{Publ. Math. Debrecen} 67(2005), 471-480.
• H. G. Nagaraja, K. Venu, Ricci solitons in Kenmotsu manifolds, \emph{Journal of Informatic and Mathematical Sciences} 8(1) (2016) 29-36.
• S. Y. Perktaş, S. Keleş, Ricci solitons in 3-dimensional normal almost paracontact metric manifolds, \emph{Int. Electron. J. Geom.} 8(2) (2015), 34-45.
• M. Petroviç, R. Roşça and L. Verstraelen, Exterior concurrent vector fields on Riemannian manifolds. I. some general results, \emph{Soochow J. Math.} 15(1989), 179-187.
• R. Sharma, Certain results on K-contact and $(k,\mu)-$contact manifolds, \emph{J. Geom.} 89, no.1-2 (2008), 138-147.
• R. Sharma, A. Ghosh, Sasakian 3-manifolds as a Ricci soliton represents the Heisenberg group, \emph{Int. J. Geom. Methods Mod. Phys} 8(1) (2011), 149-154.
• M. D. Siddiqi, M. Haseeb, M. Ahmad, Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds, \emph{Carpathian Math. Publ.} 9(2) (2017), 188-197.
• M. M. Tripathi, Ricci solitons in contact cetric manifolds, \emph{arXiv:0801.4222v1 [math DG]} (2008).
• K. Yano, Sur le parall\'{e}lisme et la concourance dans l'escape de Riemannian, \emph{Proc. Imp. Acad. Tokyo} 19(1943), 189-197.
• K. Yano, B.-Y. Chen, On the concurrent vector fields of immersed manifolds, \emph{Kodai Math. Sem. Rep.} 23(1971), 343-350.
• K. Yano, M. Kon, Structures on manifolds, Series in Mathematics, World Scientific Publishing, Springer, (1984).