| Title:
|
On the Gauss map of B-scrolls in $3$-dimensional Lorentzian space forms
(English)
|
| Author:
|
Ferrández, Angel |
| Author:
|
Lucas, Pascual |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
699-704 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note we show that $B$-scrolls over null curves in a 3-dimensional Lorentzian space form $\bar{M}^3_1(c)$ are characterized as the only ruled surfaces with null rulings whose Gauss maps $G$ satisfy the condition $\Delta G=\Lambda G$, $\Lambda \:{X}(\bar{M})\rightarrow {X}(\bar{M})$ being a parallel endomorphism of ${X}(\bar{M})$. |
| Keyword:
|
Gauss map |
| Keyword:
|
$B$-scroll |
| Keyword:
|
ruled surfce |
| MSC:
|
53C42 |
| MSC:
|
53C50 |
| idZBL:
|
Zbl 1079.53505 |
| idMR:
|
MR1792965 |
| . |
| Date available:
|
2009-09-24T10:37:13Z |
| Last updated:
|
2012-05-31 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127605 |
| . |
| Reference:
|
[AFL2-type] L. J. Alías, A. Ferrández and P. Lucas: 2-type surfaces in $\mathbb{S}_1^3$ and $\mathbb{H}_1^3$.Tokyo J. Math. 17 (1994), 447–454. MR 1305812 |
| Reference:
|
[AFLeigenvector] L. J. Alías, A. Ferrández and P. Lucas: Hypersurfaces in the non-flat Lorentzian space forms with a characteristic eigenvector field.J. Geom. 52 (1995), 10–24. MR 1317251 |
| Reference:
|
[AFLMOntheGaussmap] L. J. Alías, A. Ferrández, P. Lucas and M. A. Meroño: On the Gauss map of B-scrolls.Tsukuba J. Math. 22 (1998), 371–377. MR 1650749 |
| Reference:
|
[BBonthegauss] C. Baikoussis and D. E. Blair: On the Gauss map of ruled surfaces.Glasgow Math. J. 34 (1992), 355–359. MR 1181778 |
| Reference:
|
[ChoiGaussmap] S. M. Choi: On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space.Tsukuba J. Math. 19 (1995), 285–304. Zbl 0855.53010, MR 1366636 |
| Reference:
|
[DajczerNomizu80] M. Dajczer and K. Nomizu: On flat surfaces in $\mathbb{S}_1^3$ and $\mathbb{H}_1^3$.Manifolds and Lie Groups, Univ. Notre Dame, Indiana, Birkhäuser, 1981, pp. 71–108. MR 0642853 |
| Reference:
|
[FLpacific] A. Ferrández and P. Lucas: On surfaces in the 3-dimensional Lorentz-Minkowski space.Pacific J. Math. 152 (1992), 93–100. MR 1139974 |
| Reference:
|
[Graves79] L. Graves: Codimension one isometric immersions between Lorentz spaces.Trans. Amer. Math. Soc. 252 (1979), 367–392. Zbl 0415.53041, MR 0534127 |
| . |
