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Tubes of finite Chen-type

Hassan Al-Zoubi|Khalid M. Jaber|Stylianos Stamatakis
대한수학회 (The Korean Mathematical Society)|2018.04|대한수학회논문집 (Communications of the KMS)|제33권 제2호|pp. 581-590

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참고문헌

키워드

surfaces in the Euclidean 3-spacesurfaces of finite Chen-typeBeltrami operator
주제분류수학

초록

In this paper, we consider surfaces in the 3-dimensional Euclidean space $ \mathbb{E}^{3}$ which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in $\mathbb{E}^{3}$. We show that tubes are of infinite $III$-type.

Abstract

In this paper, we consider surfaces in the 3-dimensional Euclidean space $ \mathbb{E}^{3}$ which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in $\mathbb{E}^{3}$. We show that tubes are of infinite $III$-type.

참고문헌 (17)

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  5. B.-Y. Chen, "Some open problems and conjectures on submanifolds of finite type", Soochow J. Math., 17(2), pp.169~188 (1991)
  6. B.-Y. Chen, "A report on submanifolds of finite type", Soochow J. Math., 22(2), pp.117~337 (1996)
  7. B.-Y. Chen, "Total Mean Curvature and Submanifolds of Finite Type, 2nd edition" (2015), World Scientific Publisher
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  9. B.-Y. Chen, "Ruled surfaces of finite type", Bull. Austral. Math. Soc., 42(3), pp.447~453 (1990)[DOI]
  10. F. Defever, "The compact cyclides of Dupin and a conjecture of B.-Y. Chen", J. Geom., 46(1-2), pp.33~38 (1993)[DOI]
  11. F. Dillen, "On surfaces of finite type in Euclidean 3-space", Kodai Math. J., 13(1), pp.10~21 (1990)[DOI]
  12. W. Haack, "Elementtare Differetialgeometrie" (1955), Basel und Stuttgart
  13. O. Garay, "An extension of Takahashi's theorem", Geom. Dedicata, 34(2), pp.105~112 (1990)
  14. J. S. Ro, "Tubes of Weingarten types inEuclidean 3-space", J. Cungcheong Math. Soc., 22, pp.359~366 (2009)
  15. S. Stamatakis, "On surfaces of finite Chen-type", Results. Math., 43(1-2), pp.181~190 (2003)[DOI]
  16. S. Stamatakis, "Surfaces of revolution satisfying <TEX>${\Delta}^{III}x$<", TEX> = Ax, 2010, pp.14(2), 181~186
  17. T. Takahashi, "Minimal immersions of Riemannian manifolds", J. Math. Soc. Japan, 18, pp.380~385 (1966)[DOI]

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