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http://dx.doi.org/10.4134/JKMS.j150498

GAUSS MAPS OF RULED SUBMANIFOLDS AND APPLICATIONS I  

Jung, Sun Mi (Department of Mathematics Education Silla University)
Kim, Dong-Soo (Department of Mathematics Chonnam National University)
Kim, Young Ho (Department of Mathematics Education Kyungpook National University)
Yoon, Dae Won (Department of Mathematics Education Gyeongsang National University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1309-1330 More about this Journal
Abstract
As a generalizing certain geometric property occurred on the helicoid of 3-dimensional Euclidean space regarding the Gauss map, we study ruled submanifolds in a Euclidean space with pointwise 1-type Gauss map of the first kind. In this paper, as new examples of cylindrical ruled submanifolds in Euclidean space, we construct generalized circular cylinders and characterize such ruled submanifolds and minimal ruled submanifolds of Euclidean space with pointwise 1-type Gauss map of the first kind.
Keywords
pointwise 1-type Gauss map; ruled submanifold; Grassmannian manifold; generalized circular cylinder;
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Times Cited By KSCI : 4  (Citation Analysis)
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1 J. M. Barbosa, M. Dajczer, and L. P. Jorge, Minimal ruled submanifolds in spaces of constant curvature, Indiana Univ. Math. J. 33 (1984), no. 4, 531-547.   DOI
2 B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Second edition, World Scientific, Hackensack, NJ, 2015.
3 B.-Y. Chen, M. Choi, and Y. H. Kim, Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc. 42 (2005), no. 3, 447-455.   DOI
4 B.-Y. Chen, F. Dillen, L. Verstraelen, and L. Vrancken, Ruled surfaces of finite type, Bull. Austral. Math. Soc. 42 (1990), no. 3, 447-453.   DOI
5 B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186.   DOI
6 M. Choi, D.-S. Kim, Y. H. Kim, and D. W. Yoon, Circular cone and its Gauss map, Colloq. Math. 129 (2012), no. 2, 203-210.   DOI
7 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), no. 4, 753-761.
8 M. Choi, Y. H. Kim, and D. W. Yoon, Classification of ruled surfaces with pointwise 1-type Gauss map, Taiwanese J. Math. 14 (2010), no. 4, 1297-1308.   DOI
9 F. Dillen, Ruled submanifolds of finite type, Proc. Amer. Math. Soc. 114 (1992), no. 3, 795-798.   DOI
10 D.-S. Kim, Ruled surfaces of finite type in Lorentz space-times, Honam Math. J. 31 (2009), no. 2, 177-183.   DOI
11 D.-S. Kim, Ruled submanifolds of finite type in Lorentz space-times, Honam Math. J. 32 (2010), no. 2, 261-269.   DOI
12 D.-S. Kim and Y. H. Kim, Finite type ruled hypersurfaces in Lorentz-Minkowski space, Honam Math. J. 30 (2008), no. 4, 743-748.   DOI
13 D.-S. Kim and Y. H. Kim, Some classification results on finite type ruled submanifolds in a Lorentz-Minkowski space, Taiwanese J. Math. 16 (2012), no. 4, 1475-1488.   DOI
14 D.-S. Kim and Y. H. Kim, Minimal ruled submanifolds in Minkowski space $\mathbb{L}^m$, J. Geom. Phys. 62 (2012), no. 9, 1893-1902.   DOI
15 D.-S. Kim, Y. H. Kim, and S. M. Jung, Ruled submanifolds with Harmonic Gauss map, Taiwanese J. Math. 18 (2014), no. 1, 53-76.   DOI
16 D.-S. Kim, Y. H. Kim, and D. W. Yoon, Finite type ruled surfaces in Lorentz-Minkowski space, Taiwanese J. Math. 11 (2007), no. 1, 1-13.   DOI
17 D.-S. Kim, Y. H. Kim, and S. M. Jung, Some classifications of ruled submanifolds in Minkowski space and their Gauss map, Taiwanese J. Math. 18 (2014), no. 4, 10221-1040.
18 D.-S. Kim, Y. H. Kim, and D. W. Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math. 33 (2002), no. 7, 1031-1040.
19 D.-S. Kim, Y. H. Kim, and D. W. Yoon, Characterization of generalized B-scrolls and cylinders over finite type curves, Indian J. Pure Appl. Math. 33 (2003), no. 11, 1523-1532.
20 Y. H. Kim and D. W. Yoon, Ruled surfaces with finite type Gauss map in Minkowski spaces, Soochow J. Math. 26 (2000), no. 1, 85-96.
21 Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss maps, J. Geom. Phys. 34 (2000), no. 3-4, 191-205.   DOI
22 Y. H. Kim and D. W. Yoon, On the Gauss map of ruled surfaces in Minkowski space, Rocky Mountain J. Math. 35 (2005), no. 5, 1555-1581.   DOI