| Title:
|
Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition (English) |
| Author:
|
Lee, Hyunjin |
| Author:
|
Kim, Seonhui |
| Author:
|
Suh, Young Jin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
849-861 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $. (English) |
| Keyword:
|
real hypersurface |
| Keyword:
|
complex two-plane Grassmannians |
| Keyword:
|
Hopf hypersurface |
| Keyword:
|
commuting shape operator |
| MSC:
|
11R52 |
| MSC:
|
53C40 |
| MSC:
|
53C50 |
| MSC:
|
53C55 |
| idZBL:
|
Zbl 1265.53075 |
| idMR:
|
MR2984638 |
| DOI:
|
10.1007/s10587-012-0049-y |
| . |
| Date available:
|
2012-11-10T21:22:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143029 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/143955 |
| . |
| Reference:
|
[1] Alekseevskii, D. V.: Compact quaternion spaces.Funkts. Anal. Prilozh. 2 (1968), 11-20. MR 0231314 |
| Reference:
|
[2] Berndt, J.: Riemannian geometry of complex two-plane Grassmannian.Rend. Semin. Mat., Torino 55 (1997), 19-83. MR 1626089 |
| Reference:
|
[3] Berndt, J., Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians.Monatsh. Math. 127 (1999), 1-14. Zbl 0920.53016, MR 1666307, 10.1007/s006050050018 |
| Reference:
|
[4] Berndt, J., Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians.Monatsh. Math. 137 (2002), 87-98. Zbl 1015.53034, MR 1937621, 10.1007/s00605-001-0494-4 |
| Reference:
|
[5] Lee, H., Suh, Y. J.: Real hypersurfaces of type $B$ in complex two-plane Grassmannians related to the Reeb vector.Bull. Korean Math. Soc. 47 (2010), 551-561. Zbl 1206.53064, MR 2666376, 10.4134/BKMS.2010.47.3.551 |
| Reference:
|
[6] Pérez, J. D., Suh, Y. J.: The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians.J. Korean Math. Soc. 44 (2007), 211-235. Zbl 1156.53034, MR 2283469, 10.4134/JKMS.2007.44.1.211 |
| Reference:
|
[7] Suh, Y. J.: Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator.Bull. Aust. Math. Soc. 68 (2003), 379-393. Zbl 1058.53046, MR 2027682, 10.1017/S0004972700037795 |
| . |
