대한수학회지 (Journal of the Korean Mathematical Society)

  • 제44권2호
  • /
  • Pages.407-442
  • /
  • 2007
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

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)
  • 발행 : 2007.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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.

키워드

참고문헌

  1. C. Baikoussis, Ruled submanifolds with finite type Gauss map, J. Geom. 49 (1994), no. 1-2, 42-45 https://doi.org/10.1007/BF01228047
  2. C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359 https://doi.org/10.1017/S0017089500008946
  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 https://doi.org/10.3836/tjm/1270128488
  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 https://doi.org/10.4134/JKMS.2005.42.3.447
  11. B. Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186 https://doi.org/10.1017/S0004972700013162
  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 https://doi.org/10.2969/jmsj/02810182
  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 https://doi.org/10.2307/2045079
  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 https://doi.org/10.1016/S0393-0440(99)00063-7
  18. H. B. Jr. Lawson, Complete minimal surfaces in $S^3$, Ann. of Math. (2) 92 (1970), 335-374 https://doi.org/10.2307/1970625
  19. M. Obata, The Gauss map of immersions of Riemannian manifolds in space of constant curvature, J. Differential Geometry 2 (1968), 217-223 https://doi.org/10.4310/jdg/1214428258
  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 https://doi.org/10.2307/1995413
  22. N. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210-213 https://doi.org/10.1007/BF01220905

피인용 문헌

  1. On Submanifolds of Pseudo-Hyperbolic Space with 1-Type Pseudo-Hyperbolic Gauss Map vol.12, pp.4, 2016, https://doi.org/10.15407/mag12.04.315
  2. Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map vol.52, pp.1, 2017, https://doi.org/10.1007/s10455-017-9548-2
  3. Pseudo-Spherical Submanifolds with 1-Type Pseudo-Spherical Gauss Map vol.71, pp.3-4, 2017, https://doi.org/10.1007/s00025-016-0560-9
  4. Classification of surfaces in a pseudo-sphere with 2-type pseudo-spherical Gauss map 2017, https://doi.org/10.1002/mana.201600498
  5. Hyperbolic submanifolds with finite type hyperbolic Gauss map vol.26, pp.02, 2015, https://doi.org/10.1142/S0129167X15500147
  6. On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space vol.14, pp.1, 2017, https://doi.org/10.1007/s00009-016-0819-0