Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

Hyunjin Lee; Seonhui Kim; Young Jin Suh

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 849-861
  • ISSN: 0011-4642

Abstract

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In this paper, first we introduce a new notion of commuting condition that φ φ 1 A = A φ 1 φ between the shape operator A and the structure tensors φ and φ 1 for real hypersurfaces in G 2 ( ℂ m + 2 ) . Suprisingly, real hypersurfaces of type ( A ) , that is, a tube over a totally geodesic G 2 ( ℂ m + 1 ) in complex two plane Grassmannians G 2 ( ℂ m + 2 ) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G 2 ( ℂ m + 2 ) satisfying the commuting condition. Finally we get a characterization of Type ( A ) in terms of such commuting condition φ φ 1 A = A φ 1 φ .

How to cite

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Lee, Hyunjin, Kim, Seonhui, and Suh, Young Jin. "Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition." Czechoslovak Mathematical Journal 62.3 (2012): 849-861. <http://eudml.org/doc/247164>.

@article{Lee2012,
abstract = {In this paper, first we introduce a new notion of commuting condition that $\phi \phi _\{1\} A = A \phi _\{1\} \phi $ between the shape operator $A$ and the structure tensors $\phi $ and $\phi _\{1\}$ for real hypersurfaces in $G_2(\{\mathbb \{C\}\}^\{m+2\})$. Suprisingly, real hypersurfaces of type $(A)$, that is, a tube over a totally geodesic $G_\{2\}(\mathbb \{C\}^\{m+1\})$ in complex two plane Grassmannians $G_2(\{\mathbb \{C\}\}^\{m+2\})$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2(\{\mathbb \{C\}\}^\{m+2\})$ satisfying the commuting condition. Finally we get a characterization of Type $(A)$ in terms of such commuting condition $\phi \phi _\{1\} A = A \phi _\{1\} \phi $.},
author = {Lee, Hyunjin, Kim, Seonhui, Suh, Young Jin},
journal = {Czechoslovak Mathematical Journal},
keywords = {real hypersurface; complex two-plane Grassmannians; Hopf hypersurface; commuting shape operator; real hypersurface; complex two-plane Grassmannians; Hopf hypersurface; commuting shape operator},
language = {eng},
number = {3},
pages = {849-861},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition},
url = {http://eudml.org/doc/247164},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Lee, Hyunjin
AU - Kim, Seonhui
AU - Suh, Young Jin
TI - Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 849
EP - 861
AB - In this paper, first we introduce a new notion of commuting condition that $\phi \phi _{1} A = A \phi _{1} \phi $ between the shape operator $A$ and the structure tensors $\phi $ and $\phi _{1}$ for real hypersurfaces in $G_2({\mathbb {C}}^{m+2})$. Suprisingly, real hypersurfaces of type $(A)$, that is, a tube over a totally geodesic $G_{2}(\mathbb {C}^{m+1})$ in complex two plane Grassmannians $G_2({\mathbb {C}}^{m+2})$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2({\mathbb {C}}^{m+2})$ satisfying the commuting condition. Finally we get a characterization of Type $(A)$ in terms of such commuting condition $\phi \phi _{1} A = A \phi _{1} \phi $.
LA - eng
KW - real hypersurface; complex two-plane Grassmannians; Hopf hypersurface; commuting shape operator; real hypersurface; complex two-plane Grassmannians; Hopf hypersurface; commuting shape operator
UR - http://eudml.org/doc/247164
ER -

References

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  1. Alekseevskii, D. V., Compact quaternion spaces, Funkts. Anal. Prilozh. 2 (1968), 11-20. (1968) MR0231314
  2. Berndt, J., Riemannian geometry of complex two-plane Grassmannian, Rend. Semin. Mat., Torino 55 (1997), 19-83. (1997) MR1626089
  3. Berndt, J., Suh, Y. J., 10.1007/s006050050018, Monatsh. Math. 127 (1999), 1-14. (1999) Zbl0920.53016MR1666307DOI10.1007/s006050050018
  4. Berndt, J., Suh, Y. J., 10.1007/s00605-001-0494-4, Monatsh. Math. 137 (2002), 87-98. (2002) Zbl1015.53034MR1937621DOI10.1007/s00605-001-0494-4
  5. Lee, H., Suh, Y. J., 10.4134/BKMS.2010.47.3.551, Bull. Korean Math. Soc. 47 (2010), 551-561. (2010) Zbl1206.53064MR2666376DOI10.4134/BKMS.2010.47.3.551
  6. Pérez, J. D., Suh, Y. J., 10.4134/JKMS.2007.44.1.211, J. Korean Math. Soc. 44 (2007), 211-235. (2007) Zbl1156.53034MR2283469DOI10.4134/JKMS.2007.44.1.211
  7. Suh, Y. J., 10.1017/S0004972700037795, Bull. Aust. Math. Soc. 68 (2003), 379-393. (2003) Zbl1058.53046MR2027682DOI10.1017/S0004972700037795

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