Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

Konstantina Panagiotidou; Juan de Dios Pérez

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page Article ID 1550075, 14 p.-Article ID 1550075, 14 p.
  • ISSN: 2391-5455

Abstract

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In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.

How to cite

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Konstantina Panagiotidou, and Juan de Dios Pérez. "Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms." Open Mathematics 13.1 (2015): Article ID 1550075, 14 p.-Article ID 1550075, 14 p.. <http://eudml.org/doc/270916>.

@article{KonstantinaPanagiotidou2015,
abstract = {In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.},
author = {Konstantina Panagiotidou, Juan de Dios Pérez},
journal = {Open Mathematics},
keywords = {Real hypersurfaces; Structure Jacobi operator; Cho operator; Non-flat complex space forms; real hypersurfaces; Jacobi operators; parallelism; complex hyperbolic two-plane Grassmannians},
language = {eng},
number = {1},
pages = {Article ID 1550075, 14 p.-Article ID 1550075, 14 p.},
title = {Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms},
url = {http://eudml.org/doc/270916},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Konstantina Panagiotidou
AU - Juan de Dios Pérez
TI - Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - Article ID 1550075, 14 p.
EP - Article ID 1550075, 14 p.
AB - In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.
LA - eng
KW - Real hypersurfaces; Structure Jacobi operator; Cho operator; Non-flat complex space forms; real hypersurfaces; Jacobi operators; parallelism; complex hyperbolic two-plane Grassmannians
UR - http://eudml.org/doc/270916
ER -

References

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  1. [1] Berndt J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 1989, 395, 132-141 Zbl0655.53046
  2. [2] Cho J.T., CR-structures on real hypersurfaces of a complex space form, Publ. Math. Debrecem, 1999, 54, 473-487 Zbl0929.53029
  3. [3] Cho J.T., Pseudo-Einstein CR-structures on real hypersurfaces of a complex space form, Hokkaido Math. J., 2008, 37, 1-17 Zbl1145.53013
  4. [4] Ivey T.A., Ryan P.J., The structure Jacobi operator for real hypersurfaces in CP2 and CH2, Results Math., 2009, 56, 473-488 
  5. [5] Ivey T.A., Ryan P.J., Hopf hypersurfaces of small Hopf principal curvature in CH2, Geom. Dedicata, 2009, 141, 147-161 
  6. [6] Maeda Y., On real hypersurfaces of a complex projective space, J. Math. Soc. Japan, 1976, 28, 529-540 Zbl0324.53039
  7. [7] Montiel S., Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan, 1985, 35, 515-535 Zbl0554.53021
  8. [8] Montiel S., Romero A., On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata, 1986, 20, 245-261 Zbl0587.53052
  9. [9] Niebergall R., Ryan P.J., Real hypersurfaces in complex space forms, Math. Sci. Res. Inst. Publ., 1997, 32, 233-305 Zbl0904.53005
  10. [10] Okumura M., On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 1975, 212, 355-364 Zbl0288.53043
  11. [11] Panagiotidou K., Xenos Ph.J, Real hypersurfaces in CP2 and CH2 whose structure Jacobi operator is Lie D-parallel, Note Mat., 2012, 32, 89-99 Zbl1282.53049
  12. [12] Pérez J.D., Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space, Ann. di Mat. (in press), DOI 10.1007/s10231-014-0444-0 [WoS][Crossref] Zbl1329.53047
  13. [13] Takagi R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 1973, 10, 495-506 Zbl0274.53062
  14. [14] Tanno S., Variational problems on contact Riemannian manifolds, Trans. Am. Math. Soc., 1989, 314, 349-379 Zbl0677.53043

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