대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제32권3호
  • /
  • Pages.715-724
  • /
  • 2017
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE GAUSS MAP OF HELICOIDAL SURFACES

  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Kim, Wonyong (Department of Mathematics Chonnam National University) ;
  • Kim, Young Ho (Department of Mathematics Kyungpook National University)
  • 투고 : 2016.09.13
  • 심사 : 2016.12.22
  • 발행 : 2017.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We study the Gauss map G of helicoidal surfaces in the 3-dimensional Euclidean space ${\mathbb{E}}^3$ with respect to the so called Cheng-Yau operator ${\square}$ acting on the functions defined on the surfaces. As a result, we completely classify the helicoidal surfaces with Gauss map G satisfying ${\square}G=AG$ for some $3{\times}3$ matrix A.

키워드

참고문헌

  1. L. J. Alias and N. Gurbuz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127.
  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 and T. Koufogiorgos, Helicoidal surfaces with prescribed mean or Gaussian curvature, J. Geom. 63 (1998), no. 1-2, 25-29. https://doi.org/10.1007/BF01221235
  4. C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 2(16) (1993), 31-42.
  5. B.-Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc. 44 (1991), no. 1, 117-129. https://doi.org/10.1017/S0004972700029518
  6. 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
  7. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. https://doi.org/10.1007/BF01425237
  8. M. Choi, D.-S. Kim, and Y. H. Kim, Helicoidal surfaces with pointwise 1-type Gauss map, J. Korean Math. Soc. 46 (2009), no. 1, 215-223. https://doi.org/10.4134/JKMS.2009.46.1.215
  9. M. Choi, D.-S. Kim, Y. H. Kim, and D. W. Yoon, Circular cone and its Gauss map, Colloq. Math. 129 (2012), no. 2, 203-210. https://doi.org/10.4064/cm129-2-4
  10. S. M. Choi, On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 351-367. https://doi.org/10.21099/tkbjm/1496162874
  11. S. M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 285-304. https://doi.org/10.21099/tkbjm/1496162870
  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. M. P. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J. (2) 34 (1982), no. 3, 425-435. https://doi.org/10.2748/tmj/1178229204
  14. U. Dursun, Flat surfaces in the Euclidean space $\mathbb{E}^3$ with pointwise 1-type Gauss map, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 469-478.
  15. T. Hasanis and T. Vlachos, Hypersurfaces of $E_{n+1}\;satisfying\;{\Delta}x\;=\;Ax+B$, J. Austral. Math. Soc. Ser. A 53 (1992), no. 3, 377-384. https://doi.org/10.1017/S1446788700036545
  16. U.-H. Ki, D.-S. Kim, Y. H. Kim, and Y.-M. Roh, Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math. 13 (2009), no. 1, 317-338. https://doi.org/10.11650/twjm/1500405286
  17. D.-S. Kim, On the Gauss map of quadric hypersurfaces, J. Korean Math. Soc. 31 (1994), no. 3, 429-437.
  18. D.-S. Kim, On the Gauss map of hypersurfaces in the space form, J. Korean Math. Soc. 32 (1995), no. 3, 509-518.
  19. D.-S. Kim, J. R. Kim, and Y. H. Kim, Cheng-Yau operator and Gauss map of surfaces of revolution, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1319-1327. https://doi.org/10.1007/s40840-015-0234-x
  20. D.-S. Kim and Y. H. Kim, Surfaces with planar lines of curvature, Honam Math. J. 32 (2010), no. 4, 777-790. https://doi.org/10.5831/HMJ.2010.32.4.777
  21. D.-S. Kim, Y. H. Kim and D. W. Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math. 33 (2002), no. 7, 1031-1040.
  22. D.-S. Kim and B. Song, On the Gauss map of generalized slant cylindrical surfaces, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 20 (2013), no. 3, 149-158.
  23. Y. H. Kim and N. C. Turgay, Surfaces in $\mathbb{E}^3$ with $L_1$-pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50 (2013), no. 3, 935-949. https://doi.org/10.4134/BKMS.2013.50.3.935
  24. T. Levi-Civita, Famiglie di superficie isoparametrische nell'ordinario spacio euclideo, Atti. Accad. naz Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 26 (1937), 355-362.
  25. E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573. https://doi.org/10.1090/S0002-9947-1970-0259768-5