| Title:
|
A characterization of totally $\eta$-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form (English) |
| Author:
|
Kon, Mayuko |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1279-1287 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator $A$ of a real hypersurface $M$ of a complex space form $M^n(c)$, $c\neq 0$, $n\geq 3$, satisfies $g(AX,Y)=ag(X,Y)$ for any $X,Y\in T_0(x)$, $a$ being a function, where $T_0$ is the holomorphic distribution on $M$, then $M$ is a totally $\eta $-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of $\eta $-umbilical real hypersurfaces. (English) |
| Keyword:
|
real hypersurface |
| Keyword:
|
totally $\eta $-umbilical real hypersurface |
| Keyword:
|
ruled real hypersurface |
| MSC:
|
53C25 |
| MSC:
|
53C40 |
| MSC:
|
53C55 |
| idZBL:
|
Zbl 1174.53032 |
| idMR:
|
MR2471183 |
| . |
| Date available:
|
2010-07-21T08:19:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140457 |
| . |
| Reference:
|
[1] Cecil, T. E., Ryan, P. J.: Focal sets and real hypersurfaces in complex projective space.Trans. Am. Math. Soc. 269 (1982), 481-499. Zbl 0492.53039, MR 0637703 |
| Reference:
|
[2] Ahn, S. S., Lee, S. B., Suh, Y. J.: On ruled real hypersurfaces in a complex space form.Tsukuba J. Math. 17 (1993), 311-322. Zbl 0804.53024, MR 1255475, 10.21099/tkbjm/1496162264 |
| Reference:
|
[3] Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurface in $P^n(c)$.Math. Ann. 276 (1987), 487-497. Zbl 0605.53023, MR 0875342, 10.1007/BF01450843 |
| Reference:
|
[4] Kimura, M., Maeda, S.: On real hypersurfaces of a complex projective space.Math. Z. 202 (1989), 299-311. Zbl 0661.53015, MR 1017573, 10.1007/BF01159962 |
| Reference:
|
[5] Kon, M.: Pseudo-Einstein real hyprersurfaces in complex space forms.J. Differ. Geom. 14 (1979), 339-354. MR 0594705, 10.4310/jdg/1214435100 |
| Reference:
|
[6] Lohnherr, M., Reckziegel, H.: On ruled real hypersurfaces in complex space forms.Geom. Dedicata 74 (1999), 267-286. Zbl 0932.53018, MR 1669351, 10.1023/A:1005000122427 |
| Reference:
|
[7] Montiel, S.: Real hypersurfaces of a complex hyperbolic space.J. Math. Soc. Japan 37 (1985), 515-535. Zbl 0554.53021, MR 0792990, 10.2969/jmsj/03730515 |
| Reference:
|
[8] Ortega, M., Pérez, J. D.: Constant holomorphic sectional curvature and type number of real hypersurfaces of complex hyperbolic space.Proc. 4th Internat. Congress of Geometry, Thessaloniki, 1996 Aristotle University of Thessaloniki Thessaloniki (1997). MR 1470995 |
| Reference:
|
[9] Ortega, M., Pérez, J. D., Suh, Y. J.: Real hypersurfaces with constant totally real sectional curvature in a complex space form.Czech. Math. J. 50 (2000), 531-537. MR 1777474, 10.1023/A:1022881510000 |
| Reference:
|
[10] Sohn, D. J., Suh, Y. J.: Classification of real hypersurfaces in complex hyperbolic space in terms of constant $\varphi$-holomorphic sectional curvatures.Kyngpook Math. J. 35 (1996), 801-819. MR 1678228 |
| Reference:
|
[11] Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures.J. Math. Soc. Japan 27 (1975), 43-53. Zbl 0311.53064, MR 0355906, 10.2969/jmsj/02710043 |
| Reference:
|
[12] Tashiro, Y., Tachibana, S.: On Fubinian and $C$-Fubinian manifolds.Kōdai Math. Sem. Rep. 15 (1963), 176-183. Zbl 0116.39001, MR 0157336, 10.2996/kmj/1138844787 |
| Reference:
|
[13] Yano, K., Kon, Masahiro: $CR$ Submanifolds of Kaehlerian and Sasakian Manifolds.Birkhäuser Boston-Basel-Stuttgart (1983). Zbl 0496.53037, MR 0688816 |
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