# THREE-DIMENSIONAL ALMOST KENMOTSU MANIFOLDS WITH η-PARALLEL RICCI TENSOR

• Wang, Yaning (Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control School of Mathematics and Information Sciences Henan Normal University)
• Published : 2017.05.01
• 98 42

#### Abstract

In this paper, we prove that the Ricci tensor of a three-dimensional almost Kenmotsu manifold satisfying ${\nabla}_{\xi}h=0$, $h{\neq}0$, is ${\eta}$-parallel if and only if the manifold is locally isometric to either the Riemannian product $\mathbb{H}^2(-4){\times}\mathbb{R}$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

#### References

1. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, Volume 203, Birkhauser, 2010.
2. D. E. Blair and R. Sharma, Three dimensional locally symmetric contact metric manifolds, Boll Un. Mat. Ital. A (7) 4 (1990), no. 3, 385-390.
3. B. Cappelletti-Montano, A. D. Nicola, and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (2013), 1343002, 55 pp.
4. J. T. Cho, Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J. 45 (2016), no. 3, 435-442. https://doi.org/10.14492/hokmj/1478487619
5. J. T. Cho and M. Kimura, Reeb flow symmetry on almost contact three-manifolds, Differential Geom. Appl. 35 (2014), 266-273. https://doi.org/10.1016/j.difgeo.2014.05.002
6. J. T. Cho and J. Lee, ${\eta}$-parallel contact 3-manifolds, Bull. Korean Math. Soc. 46 (2009), no. 3, 577-589. https://doi.org/10.4134/BKMS.2009.46.3.577
7. U. C. De and G. Pathak, On 3-dimensional Kenmotsu manifolds, Indian J. Pure Appl. Math. 35 (2004), no. 2, 159-165.
8. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343-354.
9. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93 (2009), no. 1-2, 46-61. https://doi.org/10.1007/s00022-009-1974-2
10. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds with a condition of ${\eta}$-parallelism, Differential Geom. Appl. 27 (2009), no. 5, 671-679. https://doi.org/10.1016/j.difgeo.2009.03.007
11. J. Inoguchi, A note on almost contact Riemannian 3-manifolds, Bull. Yamagata Univ. Natur. Sci. 17 (2010), no. 1, 1-6.
12. D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27. https://doi.org/10.2996/kmj/1138036310
13. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594
14. J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976), no. 3, 293-329. https://doi.org/10.1016/S0001-8708(76)80002-3
15. A. M. Pastore and V. Saltarelli, Generalized nullity distributions on almost Kenmotsu manifolds, Int. Electron. J. Geom. 4 (2011), no. 2, 168-183.
16. D. Perrone, Classification of homogeneous almost cosymplectic three-manifolds, Differential Geom. Appl. 30 (2012), no. 1, 49-58. https://doi.org/10.1016/j.difgeo.2011.10.003
17. Y. Wang, Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math. 116 (2016), no. 1, 79-86.
18. Y. Wang, Ricci tensors on three-dimensional almost coKahler manifolds, Kodai Math. J. 39 (2016), no. 3, 469-483. https://doi.org/10.2996/kmj/1478073764