Previous |  Up |  Next


Gauss map; $B$-scroll; ruled surfce
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 ${X}(\bar{M})$.
[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
[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. DOI 10.1007/BF01406822 | MR 1317251
[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
[BBonthegauss] C. Baikoussis and D. E. Blair: On the Gauss map of ruled surfaces. Glasgow Math. J. 34 (1992), 355–359. DOI 10.1017/S0017089500008946 | MR 1181778
[ChoiGaussmap] S. M. Choi: On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space. Tsukuba J. Math. 19 (1995), 285–304. MR 1366636 | Zbl 0855.53010
[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
[FLpacific] A. Ferrández and P. Lucas: On surfaces in the 3-dimensional Lorentz-Minkowski space. Pacific J. Math. 152 (1992), 93–100. DOI 10.2140/pjm.1992.152.93 | MR 1139974
[Graves79] L. Graves: Codimension one isometric immersions between Lorentz spaces. Trans. Amer. Math. Soc. 252 (1979), 367–392. DOI 10.1090/S0002-9947-1979-0534127-4 | MR 0534127 | Zbl 0415.53041
Partner of
EuDML logo