## Taiwanese Journal of Mathematics

### CLASSIFICATION OF RULED SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

#### Abstract

Ruled surfaces with the Gauss map satisfying a partial differential equation which is similar to an eigenvalue problem in a 3-dimensional Euclidean space are studied. Such a Gauss map is said to be of pointwise 1-type, namely, the Gauss map $G$ satisfies $\Delta G = f(G+C)$, where $\Delta$ is the Laplacian operator, $f$ is a non-zero function and $C$ is a constant vector. As a result, such ruled surfaces are completely determined by the function $f$ and the vector $C$ when their Gauss map is of pointwise 1-type. New examples of ruled surfaces called cylinders of an infinite type and rotational ruled surfaces are introduced in this regard.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 4 (2010), 1297-1308.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405946

Digital Object Identifier
doi:10.11650/twjm/1500405946

Mathematical Reviews number (MathSciNet)
MR2663912

Zentralblatt MATH identifier
1215.53022

#### Citation

Choi, Miekyung; Kim, Young Ho; Yoon, Dae Won. CLASSIFICATION OF RULED SURFACES WITH POINTWISE 1-TYPE GAUSS MAP. Taiwanese J. Math. 14 (2010), no. 4, 1297--1308. doi:10.11650/twjm/1500405946. https://projecteuclid.org/euclid.twjm/1500405946

#### References

• C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J., 34 (1992), 355-359.
• C. Baikoussis, B.-Y. Chen and L. Verstraelen, Ruled surfaces and tubes with finite type Gauss map, Tokyo J. Math., 16 (1993), 341-348.
• B.-Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific Publ., New Jersey, 1984.
• B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117-337.
• B.-Y. Chen, M. Choi and Y. H. Kim, Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc., 42 (2005), 447-455.
• B.-Y. Chen, F. Dillen and L. Verstraelen, Finite type space curves, Soochow J. Math., 12 (1986), 1-10.
• B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 35 (1987), 161-186.
• M. Choi and Y. H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc., 38 (2001), 753-761.
• Y. H. Kim and D. W. Yoon, Ruled surfaces with finite type Gauss map in Minkowski spaces, Soochow J. Math., 26 (2000), 85-96.
• Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys., 34 (2000), 191-205.
• Y. H. Kim and D. W. Yoon, On the Gauss map of ruled surfaces in Minkowski space, Rocky Mountain J. Math., 35 (2005), 1555-1581.