Taiwanese Journal of Mathematics

SOME CLASSIFICATIONS OF RULED SUBMANIFOLDS IN MINKOWSKI SPACE AND THEIR GAUSS MAP

Dong-Soo Kim, Young Ho Kim, and Sun Mi Jung

Full-text: Open access

Abstract

Ruled submanifolds of Minkowski space with finite-type Gauss map are studied. Not having a parallel in Euclidean space, ruled submanifolds with degenerate rulings in Minkowski space drew our attention. We show that if non-cylindrical ruled submanifolds with non-degenerate rulings or ruled submanifolds with degenerate rulings have finite-type Gauss map, the Gauss map is one of the following: (1) harmonic; (2) of the so-called finite rank; (3) of null 2-type. For ruled submanifolds with degenerate rulings, we set up a relationship between finite-type immersions and immersions with finite-type Gauss map and introduce new examples of ruled submanifolds with degenerate rulings. We also characterize minimal ruled submanifolds with degenerate rulings in terms of finite-type Gauss map.

Article information

Source
Taiwanese J. Math., Volume 18, Number 4 (2014), 1021-1040.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706474

Digital Object Identifier
doi:10.11650/tjm.18.2014.3715

Mathematical Reviews number (MathSciNet)
MR3245427

Zentralblatt MATH identifier
1357.53060

Subjects
Primary: 53A04: Curves in Euclidean space 53A05: Surfaces in Euclidean space 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space

Keywords
finite type Gauss map ruled submanifold degenerate ruling non-degenerate ruling Grassmannian manifold $BS$-kind ruled submanifold $G$-kind ruled submanifold

Citation

Kim, Dong-Soo; Kim, Young Ho; Jung, Sun Mi. SOME CLASSIFICATIONS OF RULED SUBMANIFOLDS IN MINKOWSKI SPACE AND THEIR GAUSS MAP. Taiwanese J. Math. 18 (2014), no. 4, 1021--1040. doi:10.11650/tjm.18.2014.3715. https://projecteuclid.org/euclid.twjm/1499706474


Export citation

References

  • C. Baikoussis, Ruled submanifolds with finite-type Gauss map, J. Geom., 49 (1994), 42-45.
  • 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.
  • J. M. Barbosa, M. Dajczer and I. P. Jorge, Minimal ruled submanifolds in spaces of constant curvature, Indiana Univ. Math. J., 33 (1984), 531-547.
  • B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite-Type, World Scientific, Singapore, 1984.
  • B.-Y. Chen, A report on submanifolds of finite-type, Soochow J. Math., 22 (1996), 117-337.
  • B.-Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Ruled surfaces of finite-type, Bull. Austral. Math. Soc., 42 (1990), 447-453.
  • B.-Y. Chen and P. Piccinni, Submanifolds with finite-type Gauss map, Bull. Austral. Math. Soc., 35 (1987), 161-186.
  • F. Dillen, Ruled submanifolds of finite-type, Proc. Amer. Math. Soc., 114 (1992), 795-798.
  • D.-S. Kim, Ruled surfaces of finite-type in Lorentz space-times, Honam Math. J., 31 (2009), 177-183.
  • D.-S. Kim, Ruled submanifolds of finite-type in Lorentz spaec-times, Honam Math. J., 32 (2010), 261-269.
  • D.-S. Kim and Y. H. Kim, Finite-type ruled hypersurfaces in Lorentz-Minkowski space, Honam Math. J., 30 (2008), 743-748.
  • D.-S. Kim and Y. H. Kim, Some classification results on finite-type ruled submanifolds in a Lorentz-Minkowski space, Taiwan. J. Math., 16 (2012), 1475-1488.
  • D.-S. Kim and Y. H. Kim, Minimal ruled submanifolds in Minkowski space $\Bbb L^m$, J. Geom. Phys., 62 (2012), 1893-1902.
  • D.-S. Kim, Y. H. Kim and S. M. Jung, Ruled submanifolds with harmanic Gauss map, Taiwan. J. Math., to appear.
  • D.-S. Kim, Y. H. Kim and D. W. Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math., 33 (2002), 1031-1040.
  • 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), 1523-1532.
  • D.-S. Kim, Y. H. Kim and D. W. Yoon, Finite-type ruled surfaces in Lorentz-Minkowski space, Taiwan. J. Math., 11 (2007), 1-13.
  • 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, Classification of ruled surfaces in Minkowski 3-spaces, J. Geom. Phys., 49 (2004), 89-100.
  • Y. H. Kim and D. W. Yoon, On non-developable ruled surfaces in Lorentz-Minkowski 3-spaces, Taiwanese J. Math., 11 (2007), 197-214.
  • T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380-385.