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

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.

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.

Cite this paper

nullC. 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.

nullC. Machado, J. Pérez and Y. Suh, "Real Hypersurfaces in Complex Two-Plane Grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type,"

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

[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