Taiwanese Journal of Mathematics

Characterizations of Tori in $3$-spheres

Dong-Soo Kim, Young Ho Kim, and Dae Won Yoon

Full-text: Open access

Abstract

Using the $II$-metric and the $II$-Gauss map on a surface derived from the non-degenerate second fundamental form of a surface in the sphere, we establish some characterizations of compact surfaces including the spheres and the tori in the $3$-dimensional unit sphere.

Article information

Source
Taiwanese J. Math., Volume 20, Number 5 (2016), 1053-1064.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874516

Digital Object Identifier
doi:10.11650/tjm.20.2016.7247

Mathematical Reviews number (MathSciNet)
MR3555888

Zentralblatt MATH identifier
1357.53027

Subjects
Primary: 53B25: Local submanifolds [See also 53C40] 53C40: Global submanifolds [See also 53B25]

Keywords
torus principal curvatures $II$-metric finite type immersion $II$-Gauss map

Citation

Kim, Dong-Soo; Kim, Young Ho; Yoon, Dae Won. Characterizations of Tori in $3$-spheres. Taiwanese J. Math. 20 (2016), no. 5, 1053--1064. doi:10.11650/tjm.20.2016.7247. https://projecteuclid.org/euclid.twjm/1498874516


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References

  • B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics 1, World Scientific, Singapore, 1984.
  • ––––, Total Mean Curvature and Submanifolds of Finite Type, Second edition, Series in Pure Mathematics 27, World Scientific, Hackensack, NJ, 2015.
  • B.-Y. Chen and F. Dillen, Surfaces of finite type and constant curvature in the $3$-sphere, C. R. Math. Rep. Acad. Sci. Canada 12 (1990), no. 1, 47–49.
  • B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161–186.
  • M. Spivak, A Comprehensive Introduction to Differential Geometry IV, Second edition, Publish or Perish, Wilmington, Del., 1979.