Research ArticleNo Access

Ricci almost soliton and almost Yamabe soliton on Kenmotsu manifold

Amalendu Ghosh

Department of Mathematics, Chandernagore College, Hooghly, 712 136 (W.B.), India

https://doi.org/10.1142/S1793557121501308Cited by:6 (Source: Crossref)

Abstract

We prove that a Ricci almost soliton on a Kenmotsu manifold of dimension $> 3$ reduces to an expanding Ricci soliton satifying certain condition on the potential vector field or on the soliton function. Next, we show that any Ricci almost soliton on a Kenmotsu manifold is trivial (Einstein) if the soliton vector leaves the contact form $η$ invariant. Finally, we classify (locally) a Kenmotsu manifold admitting an almost Yamabe soliton. Some examples have been constructed of almost Yamabe solitons on different class of warped product spaces.

Communicated by L. W. Tu

Keywords:

AMSC: 53C25, 53C21

References

1. P. Alegre, D. E. Blair and A. Carriazo , Generalized Sasakian-space-form, Israel J. Math. 141 (2004) 157–183. Crossref, Web of Science, Google Scholar
2. A. Barros, R. Batista and E. Ribeiro Jr , Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh Math. 174(1) (2014) 29–39. Crossref, Web of Science, Google Scholar
3. A. Barros and E. Ribeiro Jr , Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140(3) (2012) 1033–1040. Crossref, Web of Science, Google Scholar
4. E. Barbosa and E. Ribeiro , On conformal solutions of the Yamabe ow, Arch. Math. 101 (2013) 79–89. Crossref, Web of Science, Google Scholar
5. A. Besse , Einstein Manifolds (Springer-Verlag, New York, 2008). Google Scholar
6. D. E. Blair , Riemannian Geometry of Contact and Symplectic Manifolds (Birkhauser, Boston, 2002). Crossref, Google Scholar
7. S. Brendle , Convergence of the Yamabe flow for arbitrary initial energy, J. Differ. Geom. 69 (2005) 217–278. Crossref, Web of Science, Google Scholar
8. E. Calvino-Louzao, M. Fernandez-Lópeź, G. García-Río and R. Vázquez-Lorenzo , Homogeneous Ricci almost solitons, Israel J. Math. 220 (2017) 531–546. Crossref, Web of Science, Google Scholar
9. H. D. Cao, X. Sun and Y. Xiaofeng , On the structure of gradient Yamabe solitons, Math. Res. Lett. 19 (2012) 767–774. Crossref, Web of Science, Google Scholar
10. B. Chow , The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Commun. Pure Appl. Math. 45 (1992) 1003–1014. Crossref, Web of Science, Google Scholar
11. P. Daskalopoulos and N. Sesum , The classification of locally conformally flat Yamabe solitons, Adv. Math. 240 (2013) 346–369. Crossref, Web of Science, Google Scholar
12. N. Ejiri , A negative answer to a conjecture of conformal transformation of Riemannian manifolds, J. Math. Soc. Japan 33(2) (1981) 261–266. Crossref, Web of Science, Google Scholar
13. A. Ghosh , Kenmotsu $3$ -metric as a Ricci soliton, Chaos, Solitons Fractals 44 (2011) 647–650. Crossref, Web of Science, Google Scholar
14. A. Ghosh , An $η$ -Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen 82(3) (2013) 591–598. Crossref, Web of Science, Google Scholar
15. A. Ghosh , Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold, Carp. Math. Publ. 11(1) (2019) 59–69. Crossref, Web of Science, Google Scholar
16. A. Ghosh , Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold, Math. Slovaca 70(1) (2020) 151–160. Crossref, Web of Science, Google Scholar
17. R. S. Hamilton, Lectures on geometric flows, unpublished (1989). Google Scholar
18. S. Y. Hsu , A note on compact gradient Yamabe solitons, J. Math. Anal. Appl. 388(2) (2012) 725–726. Crossref, Web of Science, Google Scholar
19. D. Janssens and L. Vanhecke , Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981) 1–27. Crossref, Google Scholar
20. M. Kanai , On a differential equation characterizing a Riemannian manifold, Tokyo J. Math. 6(1) (1983) 143–151. Crossref, Google Scholar
21. K. Kenmotsu , A class of almost contact Riemannian manifolds, Tôhoku Math. J. 24 (1972) 93–103. Crossref, Google Scholar
22. L. Ma and L. Cheng , Properties of complete non-compact Yamabe solitons, Ann. Glob. Anal. Geom. 40(3) (2011) 379–387. Crossref, Web of Science, Google Scholar
23. M. Okumura , On infinitesimal conformal and projective transformations of normal contact spaces, Tohoku Math. J. 14(2) (1962) 398–412. Crossref, Google Scholar
24. S. Pigola, M. Rigoli, M. Rimoldi and A. Setti , Ricci almost solitons, Ann. Scuola. Norm. Sup. Pisa. CL Sc. X(5) (2011) 757–799. Google Scholar
25. T. Seko and S. Maeta , Classification of almost Yamabe solitons in Euclidean spaces, J. Geom. Phys. 136 (2019) 97–103. Crossref, Web of Science, Google Scholar
26. S. Tanno , The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J. 21 (1969) 21–38. Crossref, Google Scholar
27. Y. Wang , Yamabe soliton on three dimensional Kenmotsu manifolds, Bull. Belg. Math. Soc. Simon Stevin 23 (2016) 345–355. Crossref, Web of Science, Google Scholar
28. K. Yano , Integral formulas in Riemannian Geometry (Marcel Dekker, New York, 1970). Google Scholar
29. R. Ye , Global existence and convergence of Yamabe flow, J. Differ. Geom. 39 (1994) 35–50. Crossref, Web of Science, Google Scholar