On Ricci Recurrent Almost Kenmotsu $3$-manifolds

Year 2024, Volume: 17 Issue: 2, 538 - 550, 27.10.2024

Ji-eun Lee

https://doi.org/10.36890/iejg.1407888

Abstract

In this paper, we prove first that for an almost Kenmotsu $3$-manifold satisfying $\xi (tr \, h^2)=0$, its Ricci operator is recurrent if and only if the manifold is locally symmetric. Next, we show that $\varphi$-Ricci symmetry and $\varphi$-Ricci recurrence are equivalent conditions in almost Kenmotsu $3$-manifolds. Thus, an almost Kenmotsu $3$-manifold is $\varphi$-Ricci symmetric if and only if it has dominantly $\eta$-parallel Ricci operator.

Keywords

Almost Kenmotsu manifold , local symmetry , Ricci recurrent

References

• [1] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhäuser Boston, Inc., Boston, second edition (2010).
• [2] Cho, J. T.: Notes on almost Kenmotsu three-manifolds. Honam Math. J. 36(3), 637–645 (2014).
• [3] Cho, J. T., Kimura, M.: Reeb flow symmetry on almost contact three-manifolds. Differential Geom. Appl. 35(suppl.), 266–273 (2014).
• [4] Dileo, G., Pastore, A. M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14(2), 343–354 (2007).
• [5] Inoguchi, J.: A note on almost contact Riemannian 3-manifolds. Bull. Yamagata Univ. Natur. Sci. 17(1), 1–6 (2010).
• [6] Inoguchi, J.: Characteristic Jacobi operator on almost Kenmotsu 3-manifolds. Int. Electron. J. Geom. 16(2), 464-525 (2023).
• [7] Inoguchi, J., Lee, J.-E.: Pseudo-symmetric almost Kenmotsu 3-manifolds. Period Math Hung (2024). https://doi.org/10.1007/s10998-024- 00591-4
• [8] Inoguchi, J., Lee, J.-E.: On the η-parallelism in almost Kenmotsu 3-manifolds. J. Korean Math. Soc. 60(6), 1303–1336 (2023).
• [9] Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4(1), 1–27 (1981).
• [10] Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972).
• [11] Olszak, Z.: On Ricci-recurrent manifolds. Colloq. Math., 52, 205-211 (1987).
• [12] Perrone, D.: Almost contact metric manifolds whose Reeb vector field is a harmonic section. Acta Math. Hung. 138, 102-126 (2013).
• [13] Perrone, D.: Differ. Geom. Appl. 59, 66–90 (2018).
• [14] Perrone, D.: Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups. Int. J. Geom. Methods Mod. Phys. 16(1) Article ID 1950011 (18 pages) (2019).
• [15] Perrone, D.: Almost contact Riemannian three-manifolds with Reeb flow symmetry. Differ. Geom. Appl. 75, Article ID 101736 11 pages (2021).
• [16] Shukla, S. S., Shukla, M. K.: On ϕ-Ricci symmetric Kenmotsu manifolds. Novi Sad J. Math. 39(2), 89–95 (2009).
• [17] Venkatesha, V., Kumara, H. A., Naik, D. M.: Ricci recurrent almost Kenmotsu 3-manifolds. Filomat 35(7), 2293–2301 (2021).