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
- A. L. Besse, Einstein Manifolds, Springer-Verlag, 2008.
- D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976.
- J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math., 30(1975), 1-41.
- 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.
- G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal., 132(2016), 66-94. https://doi.org/10.1016/j.na.2015.10.021
- 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
- T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc., 269(1982), 481-499.
- 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
- 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
- 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.
- 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
- M. Djoric and M. Okumura, CR Submanifolds of Complex Projective Space: Dev. Math. 19, Springer, New York, 2010.
- S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces: Grad. Stud. Math., 34, American Mathematical Society, Providence, RI, 2001.
- 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
- 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
- U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ., 32(1990), 207-221.
- S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II (Wiley Classics Library Ed.), A Wiley-Interscience Publ., 1996.
- M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. Differential Geom., 14(1979), 339-354.
- 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
- 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.
- 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
- 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.
- 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
- B. O'Neill, Semi-Riemannian geometry, Pure Appl. Math. Vol. 103., Academic Press, Inc., New York, 1983.
- G. Perel'man, Ricci flow with surgery on three-manifolds, math.DG/0303109, 2003.
- 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
- 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
- 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
- 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
- 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
- B. Smyth, Differential geometry of complex hypersurfaces, Ann. Math., 85(1967), 246-266. https://doi.org/10.2307/1970441
- 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
- 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
- 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
- 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
- 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
- Y. Wang, Ricci solitons on almost Kenmotsu 3-manifolds, Open Math., 15(1)(2017), 1236-1243. https://doi.org/10.1515/math-2017-0103
- Y. Wang, Ricci solitons on almost co-Kahler manifolds, Canad. Math. Bull., 62(4)(2019), 912-922. https://doi.org/10.4153/S0008439518000632
- K. Yano and M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds in CR Submanifolds, Progress in Math. 30, Birkhauser, Boston, MA., 1983.