DOI QR코드

DOI QR Code

Real Hypersurfaces in the Complex Projective Space with Pseudo Ricci-Bourguignon Solitions

  • Doo Hyun Hwang (Research Institute of Real & Complex Manifolds, Kyungpook National University) ;
  • Young Jin Suh (Department of Mathmatics & RIRCM, Kyungpook National University)
  • Received : 2023.04.06
  • Accepted : 2023.09.04
  • Published : 2024.09.30

Abstract

First, we give a complete classification of pseudo Ricci-Bourguignon soliton on real hypersurfaces in the complex projective space ℂPn = SUn+1/S(U1·Un). Next, as an application, we give a complete classification of gradient pseudo Ricci-Bourguignon soliton on real hypersurfaces in the complex projective space ℂPn.

Keywords

Acknowledgement

The first author was supported by KNU Development Project Research Fund, 2022, and the second by the Grant NRF-2021-R1C1C-2009847 from National Research Foundation of Korea.

References

  1. A. L. Besse, Einstein Manifolds, Springer-Verlag, 2008.
  2. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976.
  3. J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math., 30(1975), 1-41.
  4. J. P. Bourguignon, Ricci curvature and Einstein metrics, Global differential geometry and global analysis (Berlin, 1979) Lecture notes in Math. 838, Springer, Berlin, 42-63, 1981.
  5. G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal., 132(2016), 66-94. https://doi.org/10.1016/j.na.2015.10.021
  6. G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math., 287(2017), 337-370. https://doi.org/10.2140/pjm.2017.287.337
  7. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc., 269(1982), 481-499.
  8. P. Cernea and D. Guan, Killing fields generated by multiple solutions to the Fischer-Marsden equation, Internat. J. Math., 26(4)(2015), 1540006:1-18. https://doi.org/10.1142/S0129167X15400066
  9. S. K. Chaubey, Y. J. Suh and U.C. De, Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection, Anal. Math. Phys., 10(4)(2020), 61:1-15. https://doi.org/10.1007/s13324-019-00351-5
  10. S. K. Chaubey, U. C. De and Y. J. Suh, Kenmotsu manifolds satisfying the Fischer-Marsden equation, J. Korean Math. Soc., 58(3)(2021), 597-607.
  11. S. K. Chaubey, U. C. De and Y. J. Suh, Gradient Yamabe and gradient m-quasi Einstein metrics on three-dimensional cosymplectic manifolds, Mediterr. J. Math., 18(80(2021), 1-14. https://doi.org/10.1007/s00009-020-01619-y
  12. M. Djoric and M. Okumura, CR Submanifolds of Complex Projective Space: Dev. Math. 19, Springer, New York, 2010.
  13. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces: Grad. Stud. Math., 34, American Mathematical Society, Providence, RI, 2001.
  14. R. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986). Contemp. Math. Vol. 71., Amer. Math. Soc., Providence, RI. 237-262. https://doi.org/10.1090/conm/071/954419
  15. I. Jeong and Y. J. Suh, Pseudo anti-commuting and Ricci soiliton real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys., 86(2014), 258-272. https://doi.org/10.1016/j.geomphys.2014.08.011
  16. U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ., 32(1990), 207-221.
  17. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II (Wiley Classics Library Ed.), A Wiley-Interscience Publ., 1996.
  18. M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. Differential Geom., 14(1979), 339-354.
  19. H. Lee and Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bull. Korean Math. Soc., 47(2009), 551-561. https://doi.org/10.4134/BKMS.2010.47.3.551
  20. H. Lee and Y. J. Suh, Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric, Math. Phys. Anal. Geom. 23 (2020), no. 4, Paper No. 44, 21 pp.
  21. H. Lee and Y. J. Suh, Real hypersufaces with recurrent normal Jacobi operator in the complex quadric, J. Geom. Phys. 123 (2018), 463-474. https://doi.org/10.1016/j.geomphys.2017.10.003
  22. J. Morgan and G. Tian, Ricci flow and Poincar'e Conjecture, Clay Math. Monogr. Vol. 3., Amer. Math. Soc., Providence, RI, Clay Mathematics Institute, Cambridge, MA, 2007.
  23. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 212(1975), 355-364. https://doi.org/10.1090/S0002-9947-1975-0377787-X
  24. B. O'Neill, Semi-Riemannian geometry, Pure Appl. Math. Vol. 103., Academic Press, Inc., New York, 1983.
  25. G. Perel'man, Ricci flow with surgery on three-manifolds, math.DG/0303109, 2003.
  26. J. D. Perez, Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space, Ann. Mat. Pura Appl., 194(2015), 1781-1794. https://doi.org/10.1007/s10231-014-0444-0
  27. J. D. Perez and Y.J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. Korean Math. Soc., 44(2007), 211-235. https://doi.org/10.4134/JKMS.2007.44.1.211
  28. J. D. Perez, Y.J. Suh and Y. Watanabe, Generalized Einstein Real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys., 60(11)(2010), 1806-1818. https://doi.org/10.1016/j.geomphys.2010.06.017
  29. A. Romero, Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. Soc., 98(2)(1986), 283-286. https://doi.org/10.1090/S0002-9939-1986-0854034-6
  30. A. Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z., 192(1986), 627-635. https://doi.org/10.1007/BF01162709
  31. B. Smyth, Differential geometry of complex hypersurfaces, Ann. Math., 85(1967), 246-266. https://doi.org/10.2307/1970441
  32. Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians, Monatsh. Math., 147(2006), 337-355. https://doi.org/10.1007/s00605-005-0329-9
  33. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, J. Math. Pures Appl., 100(2013), 16-33. https://doi.org/10.1016/j.matpur.2012.10.010
  34. Y. J. Suh, Pseudo-anti commuting Ricci tensor and Ricci soliton real hypersurfaces in the complex quadric, J. Math. Pure. Appl., 107(2017), 429-450. https://doi.org/10.1016/j.matpur.2016.07.005
  35. Y. J. Suh, Real hypersurfaces in the complex quadric with Killing normal Jacobi operator, Proc. R. Soc. Edinb. A: Math., 149(2)(2019), 279-296. https://doi.org/10.1017/prm.2018.27
  36. Y. J. Suh, D. Hwang and C. Woo, Real hypersurfaces in the complex quadric with Reeb invariant Ricci tensor, J. Geom. Phys., 120(2017), 96-105. https://doi.org/10.1016/j.geomphys.2017.05.012
  37. Y. Wang, Ricci solitons on almost Kenmotsu 3-manifolds, Open Math., 15(1)(2017), 1236-1243. https://doi.org/10.1515/math-2017-0103
  38. Y. Wang, Ricci solitons on almost co-Kahler manifolds, Canad. Math. Bull., 62(4)(2019), 912-922. https://doi.org/10.4153/S0008439518000632
  39. K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds in CR Submanifolds, Progress in Math. 30, Birkhauser, Boston, MA., 1983.