Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CURVATURE HOMOGENEITY AND BALL-HOMOGENEITY ON ALMOST COKӒHLER 3-MANIFOLDS

  • Wang, Yaning (School of Mathematics and Information Sciences Henan Normal University)
  • Received : 2018.03.13
  • Accepted : 2018.06.21
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let M be a curvature homogeneous or ball-homogeneous non-$coK{\ddot{a}}hler$ almost $coK{\ddot{a}}hler$ 3-manifold. In this paper, we prove that M is locally isometric to a unimodular Lie group if and only if the Reeb vector field ${\xi}$ is an eigenvector field of the Ricci operator. To extend this result, we prove that M is homogeneous if and only if it satisfies ${\nabla}_{\xi}h=2f{\phi}h$, $f{\in}{\mathbb{R}}$.

Keywords

References

  1. D. E. Blair, The theory of quasi-Sasakian structures, J. Differential Geometry 1 (1967), 331-345. https://doi.org/10.4310/jdg/1214428097
  2. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, second edition, Progress in Mathematics, 203, Birkhauser Boston, Inc., Boston, MA, 2010.
  3. W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721-734. https://doi.org/10.2307/1970165
  4. G. Calvaruso and D. Perrone, Torsion and homogeneity on contact metric threemanifolds, Ann. Mat. Pura Appl. (4) 178 (2000), 271-285. https://doi.org/10.1007/BF02505899
  5. G. Calvaruso and D. Perrone, Natural almost contact structures and their 3D homogeneous models, Math. Nachr. 289 (2016), no. 11-12, 1370-1385. https://doi.org/10.1002/mana.201400315
  6. G. Calvaruso, D. Perrone, and L. Vanhecke, Homogeneity on three-dimensional contact metric manifolds, Israel J. Math. 114 (1999), 301-321. https://doi.org/10.1007/BF02785585
  7. G. Calvaruso and L. Vanhecke, Special ball-homogeneous spaces, Z. Anal. Anwendungen 16 (1997), no. 4, 789-800. https://doi.org/10.4171/ZAA/792
  8. B. Cappelletti-Montano, A. De Nicola, and I. Yudin, A survey on cosymplectic geometry, Rev. Math. Phys. 25 (2013), no. 10, 1343002, 55 pp. https://doi.org/10.1142/S0129055X13430022
  9. E. Garcia-Rio and L. Vanhecke, Five-dimensional $\phi$-symmetric spaces, Balkan J. Geom. Appl. 1 (1996), no. 2, 31-44.
  10. O. Kowalski and L. Vanhecke, Ball-homogeneous and disk-homogeneous Riemannian manifolds, Math. Z. 180 (1982), no. 4, 429-444. https://doi.org/10.1007/BF01214716
  11. H. Li, Topology of co-symplectic/co-Kahler manifolds, Asian J. Math. 12 (2008), no. 4, 527-543. https://doi.org/10.4310/AJM.2008.v12.n4.a7
  12. J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329. https://doi.org/10.1016/S0001-8708(76)80002-3
  13. Z. Olszak, On almost cosymplectic manifolds, Kodai Math. J. 4 (1981), no. 2, 239-250. https://doi.org/10.2996/kmj/1138036371
  14. Z. Olszak, Locally conformal almost cosymplectic manifolds, Colloq. Math. 57 (1989), no. 1, 73-87. https://doi.org/10.4064/cm-57-1-73-87
  15. 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
  16. D. Perrone, Minimal Reeb vector fields on almost cosymplectic manifolds, Kodai Math. J. 36 (2013), no. 2, 258-274. https://doi.org/10.2996/kmj/1372337517
  17. I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697. https://doi.org/10.1002/cpa.3160130408
  18. W. Wang, A class of three dimensional almost coKahler manifolds, Palest. J. Math. 6 (2017), no. 1, 111-118.
  19. 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
  20. Y. Wang, Almost co-Kahler manifolds satisfying some symmetry conditions, Turkish J. Math. 40 (2016), no. 4, 740-752. https://doi.org/10.3906/mat-1504-73
  21. K. Yamato, A characterization of locally homogeneous Riemann manifolds of dimension 3, Nagoya Math. J. 123 (1991), 77-90. https://doi.org/10.1017/S0027763000003652