Bulletin of the Korean Mathematical Society (대한수학회보)

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HOPF HYPERSURFACES OF THE HOMOGENEOUS NEARLY KÄHLER 𝕊3 × 𝕊3 SATISFYING CERTAIN COMMUTING CONDITIONS

  • Xiaomin, Chen (College of Science China University of Petroleum-Beijing) ;
  • Yifan, Yang (College of Science China University of Petroleum-Beijing)
  • Received : 2021.12.16
  • Accepted : 2022.05.02
  • Published : 2022.11.30
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Abstract

In this article, we first introduce the notion of commuting Ricci tensor and pseudo-anti commuting Ricci tensor for Hopf hypersurfaces in the homogeneous nearly Kähler 𝕊3 × 𝕊3 and prove that the mean curvature of hypersurface is constant under certain assumptions. Next, we prove the nonexistence of Ricci soliton on Hopf hypersurface with potential Reeb vector field, which improves a result of Hu et al. on the nonexistence of Einstein Hopf hypersurfaces in the homogeneous nearly Kähler 𝕊3 × 𝕊3.

Keywords

Acknowledgement

This research was supported by Science Foundation of China University of Petroleum-Beijing (Nos. 2462020XKJS02, 2462020YXZZ004). The first author expresses thanks to China Scholarship Council for supporting him to visit University of Turin and expresses his gratitude to Professor Luigi Vezzoni and Department of Mathematics for their hospitality. The authors also would like to thank the referee for the valuable comments on this paper.

References

  1. M. Antic, N. Djurdjevic, M. Moruz, and L. Vrancken, Three-dimensional CR submanifolds of the nearly Kahler 𝕊3×𝕊3, Ann. Mat. Pura Appl. (4) 198 (2019), no. 1, 227-242. https://doi.org/10.1007/s10231-018-0770-8
  2. B. Bekta,s, M. Moruz, J. Van der Veken, and L. Vrancken, Lagrangian submanifolds with constant angle functions of the nearly Kahler 𝕊3×𝕊3, J. Geom. Phys. 127 (2018), 1-13. https://doi.org/10.1016/j.geomphys.2018.01.011
  3. B. Bekta,s, M. Moruz, J. Van der Veken, and L. Vrancken, Lagrangian submanifolds of the nearly Kahler 𝕊3×𝕊3 from minimal surfaces in 𝕊3, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 3, 655-689. https://doi.org/10.1017/prm.2018.43
  4. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex twoplane Grassmannians, Monatsh. Math. 137 (2002), no. 2, 87-98. https://doi.org/10.1007/s00605-001-0494-4
  5. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, Internat. J. Math. 24 (2013), no. 7, 1350050, 18 pp. https://doi.org/10.1142/S0129167X1350050X
  6. J.-B. Butruille, Homogeneous nearly Kahler manifolds, in Handbook of pseudo-Riemannian geometry and supersymmetry, 399-423, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zurich, 2010. https://doi.org/10.4171/079-1/11
  7. J. Bolton, F. Dillen, B. Dioos, and L. Vrancken, Almost complex surfaces in the nearly Kahler 𝕊3×𝕊3, Tohoku Math. J. (2) 67 (2015), no. 1, 1-17. https://doi.org/10.2748/tmj/1429549576
  8. J. T. Cho, Notes on contact Ricci solitons, Proc. Edinb. Math. Soc. (2) 54 (2011), no. 1, 47-53. https://doi.org/10.1017/S0013091509000571
  9. B. Dioos, L. Vrancken, and X. Wang, Lagrangian submanifolds in the homogeneous nearly Kahler 𝕊3×𝕊3, Ann. Global Anal. Geom. 53 (2018), no. 1, 39-66. https://doi.org/10.1007/s10455-017-9567-z
  10. M. Djoric, M. Djoric, and M. Moruz, Real hypersurfaces of the homogeneous nearly Kahler 𝕊3×𝕊3 with P-isotropic normal, J. Geom. Phys. 160 (2021), Paper No. 103945, 13 pp. https://doi.org/10.1016/j.geomphys.2020.103945
  11. L. Foscolo and M. Haskins, New G2-holonomy cones and exotic nearly Kahler structures on 𝕊6 and 𝕊3×𝕊3, Ann. of Math. (2) 185 (2017), no. 1, 59-130. https://doi.org/10.4007/annals.2017.185.1.2
  12. Z. Hu, M. Moruz, L. Vrancken, and Z. Yao, On the nonexistence and rigidity for hypersurfaces of the homogeneous nearly Kahler 𝕊3×𝕊3, Differential Geom. Appl. 75 (2021), Paper No. 101717, 22 pp. https://doi.org/10.1016/j.difgeo.2021.101717
  13. Z. Hu and Z. Yao, On Hopf hypersurfaces of the homogeneous nearly Kahler 𝕊3×𝕊3, Ann. Mat. Pura Appl. (4) 199 (2020), no. 3, 1147-1170. https://doi.org/10.1007/s10231-019-00915-z
  14. Z. Hu, Z. Yao, and X. Zhang, Hypersurfaces of the homogeneous nearly Kahler 𝕊6 and 𝕊3×𝕊3 with anticommutative structure tensors, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 4, 535-549. https://doi.org/10.36045/bbms/1576206356
  15. Z. Hu, Z. Yao, and Y. Zhang, On some hypersurfaces of the homogeneous nearly Kahler 𝕊3×𝕊3, Math. Nachr. 291 (2018), no. 2-3, 343-373. https://doi.org/10.1002/mana.201600398
  16. Z. Hu and Y. Zhang, Rigidity of the almost complex surfaces in the nearly Kahler 𝕊3×𝕊3, J. Geom. Phys. 100 (2016), 80-91. https://doi.org/10.1016/j.geomphys.2015.10.008
  17. Z. Hu and Y. Zhang, Isotropic Lagrangian submanifolds in the homogeneous nearly Kahler 𝕊3×𝕊3, Sci. China Math. 60 (2017), no. 4, 671-684. https://doi.org/10.1007/s11425-016-0288-0
  18. I. Jeong and Y. J. Suh, Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys. 86 (2014), 258-272. https://doi.org/10.1016/j.geomphys.2014.08.011
  19. P.-A. Nagy, Nearly Kahler geometry and Riemannian foliations, Asian J. Math. 6 (2002), no. 3, 481-504. https://doi.org/10.4310/AJM.2002.v6.n3.a5
  20. Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians, Monatsh. Math. 147 (2006), no. 4, 337-355. https://doi.org/10.1007/s00605-005-0329-9
  21. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor, J. Geom. Phys. 60 (2010), no. 11, 1792-1805. https://doi.org/10.1016/j.geomphys.2010.06.007
  22. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with ξ-invariant Ricci tensor, J. Geom. Phys. 61 (2011), no. 4, 808-814. https://doi.org/10.1016/j.geomphys.2010.12.010
  23. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1309-1324. https://doi.org/10.1017/S0308210510001472
  24. Y. J. Suh, Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor, J. Geom. Phys. 106 (2016), 130-142. https://doi.org/10.1016/j.geomphys.2016.03.001
  25. Y. J. Suh, Pseudo-anti commuting Ricci tensor and Ricci soliton real hypersurfaces in the complex quadric, J. Math. Pures Appl. (9) 107 (2017), no. 4, 429-450. https://doi.org/10.1016/j.matpur.2016.07.005