Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 989-1004, 2017
Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators
Doo Hyun Hwang, Eunmi Pak, Changhwa Woo
Received June 7, 2016. First published August 15, 2017.
Abstract: We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians ${\rm SU}_{2,m}/S(U_2{\cdot}U_m)$ with commuting conditions between the restricted normal Jacobi operator $\overline{R}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$).
Keywords: real hypersurface; complex hyperbolic two-plane Grassmannians; Hopf hypersurface; shape operator; Ricci tensor; normal Jacobi operator; commuting condition
References: [1] L. J. Alías, A. Romero, M. Sánchez: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. Gen. Relativ. Gravitation 27 (1995), 71-84. DOI 10.1007/BF02105675 | MR 1310212 | Zbl 0908.53034 [2] L. J. Alías, A. Romero, M. Sánchez: Spacelike hypersurfaces of constant mean curvature in spacetimes with symmetries. Proc. Conf., Valencia, 1998 (E. Llinares Fuster et al., eds.). Publ. R. Soc. Mat. Esp. 1, Real Sociedad Matemática Española, Madrid (2000), 1-14. MR 1791221 | Zbl 0984.53025 [3] J. Berndt, Y. J. Suh: Contact hypersurfaces in Kähler manifolds. Proc. Am. Math. Soc. 143 (2015), 2637-2649. DOI 10.1090/S0002-9939-2015-12421-5 | MR 3326043 | Zbl 1318.53052 [4] J. M. Latorre, A. Romero: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. Geom. Dedicata 93 (2002), 1-10. DOI 10.1023/A:1020341512060 | MR 1934681 | Zbl 1029.53072 [5] H. Lee, Y. J. Suh, C. Woo: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators. Mediterr. J. Math. 13 (2016), 3389-3407. DOI 10.1007/s00009-016-0692-x | MR 3554315 | Zbl 06661842 [6] J. D. Pérez, Y. J. Suh, C. Woo: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator. Open Math. 13 (2015), 493-501. DOI 10.1515/math-2015-0046 | MR 3391385 | Zbl 1348.53064 [7] Y. J. Suh: Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians. Adv. Appl. Math. 50 (2013), 645-659. DOI 10.1016/j.aam.2013.01.001 | MR 3032310 | Zbl 1279.53051 [8] Y. J. Suh: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field. Adv. Appl. Math. 55 (2014), 131-145. DOI 10.1016/j.aam.2014.01.005 | MR 3176718 | Zbl 1296.53123 [9] Y. J. Suh: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting Ricci tensor. Int. J. Math. 26 (2015), Article ID 1550008, 26 pages. DOI 10.1142/S0129167X15500081 | MR 3313653 | Zbl 1335.53075 [10] Y. J. Suh: Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math. 281 (2015), 886-905. DOI 10.1016/j.aim.2015.05.012 | MR 3366856 | Zbl 06458142 [11] Y. J. Suh, C. Woo: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor. Math. Nachr. 287 (2014), 1524-1529. DOI 10.1002/mana.201300283 | MR 3256976 | Zbl 1307.53043
Affiliations: Doo Hyun Hwang, Eunmi Pak, Department of Mathematics, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 702-701, Republic of Korea, e-mail: engus0322@knu.ac.kr, empak@knu.ac.kr; Changhwa Woo, Department of Mathematics Education, Woosuk University, 443 Samnyero, Samnye, 565-701 Wanju, Jeonbuk, Republic of Korea, e-mail: legalgwch@naver.com
