Browse > Article
http://dx.doi.org/10.4134/JKMS.j150416

GRADIENT RICCI ALMOST SOLITONS ON TWO CLASSES OF ALMOST KENMOTSU MANIFOLDS  

Wang, Yaning (Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control School of Mathematics and Information Sciences Henan Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1101-1114 More about this Journal
Abstract
Let ($M^{2n+1}$, ${\phi}$, ${\xi}$, ${\eta}$, g) be a (k, ${\mu}$)'-almost Kenmotsu manifold with k < -1 which admits a gradient Ricci almost soliton (g, f, ${\lambda}$), where ${\lambda}$ is the soliton function and f is the potential function. In this paper, it is proved that ${\lambda}$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton ${\mathbb{H}}^{n+1}(-4){\times}{\mathbb{R}}^n$, and the soliton is expanding with ${\lambda}=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature -1 or the potential vector field is pointwise colinear with the Reeb vector field.
Keywords
gradient Ricci almost soliton; (k, ${\mu}$)'-almost Kenmotsu manifold; 3-dimensional Kenmotsu manifold; Einstein metric;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. Barros, R. Batista, and E. Jr. Ribeiro, Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh. Math. 174 (2014), no. 1, 29-39.   DOI
2 D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, Volume 203, Birkhauser, 2010.
3 C. P. Boyer and K. Galicki, Einstein manifolds and contact geometry, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2419-2430.   DOI
4 C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 361-368.
5 J. T. Cho, Almost contact 3-manifolds and Ricci solitons, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 1, 1220022, 7 pages.   DOI
6 J. T. Cho and R. Sharma, Contact geometry and Ricci solitons, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 6, 951-960.   DOI
7 G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343-354.
8 G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93 (2009), no. 1-2, 46-61.   DOI
9 A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons Fractals 44 (2011), no. 8, 647-650.   DOI
10 A. Ghosh, An $\eta$-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen 82 (2013), no. 3-4, 691-598.
11 A. Ghosh, Certain contact metric as Ricci almost solitons, Results Math. 65 (2014), no. 1-2, 81-94.   DOI
12 R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306.   DOI
13 R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
14 D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27.   DOI
15 K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972), no. 1, 93-103.   DOI
16 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint, http://arXiv.org/abs/math.DG/0211159.
17 P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2009), no. 2, 329-345.   DOI
18 P. Petersen and W. Wylie, On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2085-2092.   DOI
19 S. Pigola, M. Rigoli, M. Rimoldi, and M. Setti, Ricci almost solitons, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 10, (2011), no. 4, 757-799.
20 R. Sharma, Certain results on K-contact and (k, ${\mu}$)-contact manifolds, J. Geom. 89 (2008), no. 1-2, 138-147.   DOI
21 R. Sharma, Almost Ricci solitons and K-contact geometry, Monatsh. Math. 175 (2014), no. 4, 621-628.   DOI
22 Y. Wang, U. C. De, and X. Liu, Gradient Ricci solitons on almost Kenmotsu manifolds, Publ. Inst. Math. 98 (2015), no. 112, 227-235.   DOI
23 Y. Wang and X. Liu, Ricci solitons on three-dimensional $\eta$-Einstein almost Kenmotsu manifolds, Taiwanese J. Math. 19 (2015), no. 1, 91-100.   DOI
24 Y. Wang and X. Liu, Locally symmetric CR-integrable almost Kenmotsu manifolds, Mediterr. J. Math. 12 (2015), no. 1, 159-171.   DOI
25 Y. Wang and X. Liu, On almost Kenmotsu manifolds satisfying some nullity distributions, submitted.
26 K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.