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Real Hypersurfaces in Complex Two-Plane Grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type

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DOI: 10.4236/apm.2011.13015    4,502 Downloads   9,429 Views   Citations

ABSTRACT

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi operators or the Jacobi corresponding to the directions in the distribution are of Codazzi type if they satisfy a further condition. We obtain that that they must be either of type (A) or of type (B) (see [2]), but no one of these satisfies our condition. As a consequence, we obtain the non-existence of Hopf real hypersurfaces in such ambient spaces whose Jacobi operators corresponding to -directions are parallel with the same further condition.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

C. Machado, J. Pérez and Y. Suh, "Real Hypersurfaces in Complex Two-Plane Grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type," Advances in Pure Mathematics, Vol. 1 No. 3, 2011, pp. 67-72. doi: 10.4236/apm.2011.13015.

References

[1] J. Berndt, S. Console and C. Olmos, “Submanifolds and Holonomy,” Chapman & Hall CRC, Research Notes in Mathematics, Boca Raton, Vol. 434, 2003.
[2] J. Berndt and Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians,” Monatshefte für Mathematik, Vol. 127, No. 1, 1999, pp. 1-14. doi:10.1007/s006050050018
[3] J. Berndt and Y.-J. Suh, “Real Hypersurfaces with Isometric Reeb Flow on Real Hypersurfaces in Complex Two-Plane Grassmannians,” Monatshefte für Mathematik, Vol. 137, No. 2, 2002, pp. 87-98. doi:10.1007/s00605-001-0494-4
[4] I. Jeong, H.-J. Kim and Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Parallel Normal Jacobi Operator,” Publicationes Mathematicae Debrecen, Vol. 76, No. 1-2, 2010, pp. 203-218.
[5] I. Jeong, C. J. G. Machado, J. D. Pérez and Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with -Parallel Structure Jacobi Operator,” International Journal of Mathematics, Vol. 22, 2011.
[6] I. Jeong, J. D. Pérez and Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Commuting Normal Jacobi Operator,” Acta Mathematica Hungarica, Vol. 117, No. 3, 2007, pp. 201-217. doi:10.1007/s10474-007-6091-9
[7] I. Jeong and Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Lie -Parallel Normal Jacobi Operator,” Journal of the Korean Mathematical Society, Vol. 45, No. 4, 2008, pp. 1113-1133. doi:10.4134/JKMS.2008.45.4.1113
[8] M. Kimura, “Real Hypersurfaces and Complex Submanifolds in Complex Projective Space,” Transactions of the American Mathematical Society, Vol. 296, No. 1, 1986, pp. 137-149. doi:10.1090/S0002-9947-1986-0837803-2
[9] H.-J. Lee and Y.-J. Suh, “Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians Related to the Reeb Vector,” Bulletin of the Korean Mathematical Society, Vol. 47, No. 3, 2010, pp. 551-561. doi:10.4134/BKMS.2010.47.3.551
[10] A. Martinez and J. D. Pérez, “Real Hypersurfaces in Quaternionic Projective Space,” Annali di Matematica Pura ed Applicata, Vol. 145, No. 1, 1986, pp. 355-384. doi:10.1007/BF01790548
[11] J. D. Pérez and Y.-J. Suh, “The Ricci Tensor of Real Hypersurfaces in Complex Two-Plane Grassmannians,” Journal of the Korean Mathematical Society, Vol. 44, No. 1, 2007, pp. 211-235.
[12] Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Parallel Shape Operator,” Bulletin of the Australian Mathematical Society, Vol. 67, 2003, pp. 493-502. doi:10.1017/S000497270003728X
[13] Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Parallel Shape Operator II,” Journal of the Korean Mathematical Society, Vol. 41, No. 3, 2004, pp. 535-565.
[14] Y.-J. Suh, “Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivatives,” Canadian Math. Bull., Vol. 49, No. 1, 2006, pp. 134-143. doi:10.4153/CMB-2006-014-8
[15] Y.-J. Suh, “Real Hypersurfaces of Type in Complex Two-Plane Grassmannians,” Monatshefte für Mathematik, Vol. 147, No. 4, 2006, pp. 337-355. doi:10.1007/s00605-005-0329-9

  
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