Real Hypersurfaces in Complex Two-plane Grassmannians with F-parallel Normal Jacobi Operator

• Journal title : Kyungpook mathematical journal
• Volume 51, Issue 4,  2011, pp.395-410
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2011.51.4.395
Title & Authors
Real Hypersurfaces in Complex Two-plane Grassmannians with F-parallel Normal Jacobi Operator
Jeong, Im-Soon; Suh, Young-Jin;

Abstract
In this paper we give a non-existence theorem for Hopf hypersurfaces M in complex two-plane Grassmannians $\small{G_2(\mathbb{C}^{m+2})}$ whose normal Jacobi operator $\small{\bar{R}_N}$ is parallel on the distribution F defined by $F Keywords Real hypersurfaces;Complex two-plane Grassmannians;F-parallel normal Jacobi operator;Hopf hypersurfaces; Language English Cited by 1. Characterizations for Real Hypersurfaces in Complex and Quaternionic Space Forms Related to the Normal Jacobi Operator,;;; Kyungpook mathematical journal, 2016. vol.56. 3, pp.993-1002 1. Characterizations for Real Hypersurfaces in Complex and Quaternionic Space Forms Related to the Normal Jacobi Operator, Kyungpook mathematical journal, 2016, 56, 3, 993 2. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator, Czechoslovak Mathematical Journal, 2015, 65, 1, 207 3. Generalized Tanaka-Webster and Levi-Civita connections for normal Jacobi operator in complex two-plane Grassmannians, Czechoslovak Mathematical Journal, 2015, 65, 2, 569 4. Semi-parallelism of normal Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians, Monatshefte für Mathematik, 2013, 172, 2, 167 5. Commuting structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians, Acta Mathematica Sinica, English Series, 2015, 31, 1, 111 References 1. D. V. Alekseevskii, Compact quaternion spaces, Func. Anal. Appl., 2(1968), 106-114. 2. J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math., 419(1991), 9-26. 3. J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte fur Math., 127(1999), 1-14. 4. J. Berndt and Y. J. Suh, Isometric ows on real hypersurfaces in complex two-plane Grassmannians, Monatshefte fur Math., 137(2002), 87-98. 5. I. Jeong, J. D. Perez and Y. J. Suh Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator, Acta Math. Hungarica, 117(2007), 201-217. 6. I. Jeong and Y. J. Suh Real hypersurfaces in complex two-plane Grassmannians with Lie$\xi$-parallel normal Jacobi operator, J. of Korean Math. Soc., 45(2008), 1113-1133. 7. I. Jeong, H. J. Kim and Y. J. Suh Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator, Publ. Math. Debrecen, 76(2010), 203-218. 8. U-H. Ki, J. D. Perez, F. G. Santos and Y. J. Suh Real hypersurfaces in complex space forms with$\xi$-parallel Ricci tensor and structure Jacobi operator, J. of Korean Math. Soc., 44(2007), 307-326. 9. U-H. Ki and Y. J. Suh, Real hypersurfaces in complex hyperbolic space with commuting Ricci tensor, Kyungpook Math. J., 48(2008), 433-442. 10. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296(1986), 137-149. 11. J. D. Perez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying${\nabla}_U_{\iota}R$= 0, Diff. Geom. and Its Appl., 7(1997), 211-217. 12. J. D. Perez and Y. J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. of Korean Math. Soc., 44(2007), 211-235. 13. J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie$\xi\$-parallel, Diff. Geom. and Its Appl., 22(2005), 181-188.

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