대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제50권5호
  • /
  • Pages.1599-1622
  • /
  • 2013
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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CURVES ON THE UNIT 3-SPHERE S3(1) IN EUCLIDEAN 4-SPACE ℝ4

  • 투고 : 2012.09.07
  • 발행 : 2013.09.30
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초록

We show many examples of curves on the unit 2-sphere $S^2(1)$ in $\mathbb{R}^3$ and the unit 3-sphere $S^3(1)$ in $\mathbb{R}^4$. We study whether its curves are Bertrand curves or spherical Bertrand curves and provide some examples illustrating the resultant curves.

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참고문헌

  1. Yu. Aminov, Differential Geometry and Topology of Curves, Gordon and Breach Science Publishers, Amsterdam, 2000.
  2. Yu. Aminov, The Geometry of Submanifolds, Gordon and Breach Science Publishers, Amsterdam, 2001.
  3. H. Balgetir, M. Bektas, and J.-I. Inoguchi, Null Bertrand curves in Minkowski 3-space and their characterizations, Note Mat. 23 (2004), no. 1, 7-13.
  4. P. A. Blaga, Lectures on the Differential Geometry of Curves and Surfaces, Napoca Press, 2005.
  5. M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ. 1976.
  6. N. Ekmekci and K. Ilarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.
  7. C.-C. Hsiung, A First Course in Differential Geometry, International Press, Cambridge, MA. 1997.
  8. S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom. 74 (2002), no. 1-2, 97-109. https://doi.org/10.1007/PL00012543
  9. P. Lucas and J. A. Ortega-Yagues, Bertrand curves in the three-dimensional sphere, J. Geom. Phys. 62 (2012), no. 9, 1903-1914. https://doi.org/10.1016/j.geomphys.2012.04.007
  10. H. Matsuda and S. Yorozu, Notes on Bertrand curves, Yokohama Math. J. 50 (2003), no. 1-2, 41-58.
  11. R. S. Millman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc, 1977.
  12. J. Monterde, Curves with constant curvature ratios, Bol. Soc. Mat. Mexicana (3) 13 (2007), no. 1, 177-186.
  13. L. R. Pears, Bertrand curves in Riemannian space, J. London Math. Soc. (1935), no. 2, 180-183.
  14. W. K. Schief, On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys. 45 (2003), no. 1-2, 130-150. https://doi.org/10.1016/S0393-0440(02)00130-4