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SOME CLASSIFICATIONS OF RULED SUBMANIFOLDS

Kim, Dong-Soo;Kim, Young Ho;Jung, Sun Mi

  • Received : 2013.04.23
  • Published : 2014.05.31

Abstract

Ruled submanifolds in Euclidean space satisfying some algebraic equations concerning the Laplace operator related to the isometric immersion and Gauss map are studied. Cylinders over a finite type curve or generalized helicoids are characterized with such algebraic equations.

Keywords

Gauss map;ruled submanifold;Grassmannian manifold

References

  1. B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.
  2. C. Baikoussis, Ruled submanifolds with finite type Gauss map, J. Geom. 49 (1994), no. 1-2, 42-45. https://doi.org/10.1007/BF01228047
  3. C. Baikoussis and D. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359. https://doi.org/10.1017/S0017089500008946
  4. J. M. Barbosa, M. Dajczer, and I. P. Jorge, Minimal ruled submanifolds in spaces of constant curvature, Indiana Univ. Math. J. 33 (1984), no. 4, 531-547. https://doi.org/10.1512/iumj.1984.33.33028
  5. B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117-337.
  6. B.-Y. Chen and M. Petrovic, On spectral decomposition of immersion of finite type, Bull. Austral. Math. Soc. 44 (1991), no. 2, 117-129. https://doi.org/10.1017/S0004972700029518
  7. F. Dillen, Ruled submanifolds of finite type, Proc. Amer. Math. Soc. 114 (1992), no. 3, 795-798. https://doi.org/10.1090/S0002-9939-1992-1072333-5
  8. 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.
  9. D.-S. Kim, On the Gauss map of quadric hypersurfaces, J. Korean Math. Soc. 31 (1994), no. 3, 429-437.
  10. D.-S. Kim and H.-S. Chung, Space curves satisfying ${\Delta}H=AH$, Bull. Korean Math. Soc. 31 (1994), no. 2, 193-200.
  11. U. Lumiste, Die n-dimensionalen Minimalflachen mit eineer (n + 1)-dimensionalen asymtotischen Richitung im jeden Punkute, Tarrtu Riikl. Ul. Toimetised 62 (1958), 117-141.
  12. T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. https://doi.org/10.2969/jmsj/01840380