JOURNAL BROWSE
Search
Advanced SearchSearch Tips
SPHERICAL SUBMANIFOLDS WITH FINITE TYPE SPHERICAL GAUSS MAP
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
SPHERICAL SUBMANIFOLDS WITH FINITE TYPE SPHERICAL GAUSS MAP
Chen, Bang-Yen; Lue, Huei-Shyong;
  PDF(new window)
 Abstract
The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.
 Keywords
spherical Gauss map;finite type map;Clifford minimal torus;Veronese surface;equilateral torus;
 Language
English
 Cited by
1.
Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map, Results in Mathematics, 2016  crossref(new windwow)
2.
Hyperbolic submanifolds with finite type hyperbolic Gauss map, International Journal of Mathematics, 2015, 26, 02, 1550014  crossref(new windwow)
 References
1.
C. Baikoussis, Ruled submanifolds with finite type Gauss map, J. Geom. 49 (1994), no. 1-2, 42-45 crossref(new window)

2.
C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359 crossref(new window)

3.
C. Baikoussis, B. Y. Chen and L. Verstraelen, Ruled surfaces and tubes with finite type Gauss map, Tokyo J. Math. 16 (1993), no. 2, 341-349 crossref(new window)

4.
C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 16 (1993), 31-42

5.
B. Y. Chen, Geometry of submanifolds, Pure and Applied Mathematics, No. 22, Mercer Dekker, New York, 1973

6.
B. Y. Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, 1, World Scientific, New Jersey, 1984

7.
B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337

8.
B. Y. Chen, Riemannian Submanifolds, in Handbook of Differential Geometry, vol. I, North Holland, (edited by F. Dillen and L. Verstraelen) 2000, pp. 187-418

9.
B. Y. Chen and S. J. Li, Spherical hypersurfaces with 2-type Gauss map, Beitrage Algebra Geom. 39 (1998), no. 1, 169-179

10.
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 crossref(new window)

11.
B. Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186 crossref(new window)

12.
F. Dillen, J. Pas and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990), no. 3, 239-246

13.
K. O. Jang and Y. H. Kim, 2-type surfaces with 1-type Gauss map, Commun. Korean Math. Soc. 12 (1997), no. 1, 79-86

14.
K. Kenmotsu, On minimal immersions of $R^2$ into $S^n$, J. Math. Soc. Japan 28 (1976), no. 1, 182-191 crossref(new window)

15.
K. Kenmotsu, On Veronese-Boruvka spheres, Arch. Math. (Brno) 33 (1997), no. 1-2, 37-40

16.
K. Kenmotsu, Minimal surfaces with constant curvature in 4-dimensional space forms, Proc. Amer. Math. Soc. 89 (1983), no. 1, 133-138 crossref(new window)

17.
Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), no. 3-4, 191-205 crossref(new window)

18.
H. B. Jr. Lawson, Complete minimal surfaces in $S^3$, Ann. of Math. (2) 92 (1970), 335-374 crossref(new window)

19.
M. Obata, The Gauss map of immersions of Riemannian manifolds in space of constant curvature, J. Differential Geometry 2 (1968), 217-223

20.
R. Osserman, Minimal surfaces, Gauss maps, total curvature, eigenvalues estimates and stability, in: The Chern Symposium, pp. 199-227, Berkeley, 1979

21.
E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573 crossref(new window)

22.
N. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210-213 crossref(new window)