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http://dx.doi.org/10.4134/JKMS.2007.44.2.407

SPHERICAL SUBMANIFOLDS WITH FINITE TYPE SPHERICAL GAUSS MAP  

Chen, Bang-Yen (Department of Mathematics Michigan State University)
Lue, Huei-Shyong (Department of Computer Sciences and Information Engineering Yuanpei Institute of Science and Technology Hsinchu)
Publication Information
Journal of the Korean Mathematical Society / v.44, no.2, 2007 , pp. 407-442 More about this Journal
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 $S^{n+1}$ 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;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 1  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
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1 K. Kenmotsu, On minimal immersions of $R^2$ into $S^n$, J. Math. Soc. Japan 28 (1976), no. 1, 182-191   DOI
2 N. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210-213   DOI
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   DOI
4 B. Y. Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, 1, World Scientific, New Jersey, 1984
5 K. Kenmotsu, On Veronese-Boruvka spheres, Arch. Math. (Brno) 33 (1997), no. 1-2, 37-40
6 C. Baikoussis, Ruled submanifolds with finite type Gauss map, J. Geom. 49 (1994), no. 1-2, 42-45   DOI
7 C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359   DOI
8 C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 16 (1993), 31-42
9 B. Y. Chen, Geometry of submanifolds, Pure and Applied Mathematics, No. 22, Mercer Dekker, New York, 1973
10 B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337
11 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
12 B. Y. Chen and S. J. Li, Spherical hypersurfaces with 2-type Gauss map, Beitrage Algebra Geom. 39 (1998), no. 1, 169-179
13 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   ScienceOn
14 B. Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186   DOI
15 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
16 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
17 K. Kenmotsu, Minimal surfaces with constant curvature in 4-dimensional space forms, Proc. Amer. Math. Soc. 89 (1983), no. 1, 133-138   DOI   ScienceOn
18 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   DOI   ScienceOn
19 H. B. Jr. Lawson, Complete minimal surfaces in $S^3$, Ann. of Math. (2) 92 (1970), 335-374   DOI
20 M. Obata, The Gauss map of immersions of Riemannian manifolds in space of constant curvature, J. Differential Geometry 2 (1968), 217-223   DOI
21 E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573   DOI
22 R. Osserman, Minimal surfaces, Gauss maps, total curvature, eigenvalues estimates and stability, in: The Chern Symposium, pp. 199-227, Berkeley, 1979