ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
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A characterization of ccr-curves in Rm; pp. 217–224

Full article in PDF format | doi: 10.3176/proc.2008.4.03

Authors
Günay Öztürk, Kadri Arslan, H. Hilmi Hacisalihoglu

Abstract

We study the curve in Rm  for which the ratios between two consecutive curvatures are constant (ccr-curves). We show that closed ccr-curves in Euclidean space Rm  are of finite type. We also consider Frenet curves with constant harmonic curvatures and show that an immersed curve in R2n+1   with constant harmonic curvatures Hi at point γ (s0)   has a Darboux vertex at that point.


References

  1. Arslan, K., Celik, Y., and Hacısalihoğlu, H. H. On harmonic curvatures of a Frenet curve. Common. Fac. Sci. Univ. Ank. Series AV 1}, 2000, 49, 15–23.

  2. Chen, B. Y. A report on submanifolds of finite type. Soochow J. Math., 1996, 22, 117–337.

  3. Chen, B. Y. On submanifolds of finite type. Soochow J. Math., 1983, 9, 65–81.

  4. Chen, B. Y. On the total curvature of immersed manifolds, VI: Submanifolds of finite type and their applications. Bull. Inst. Math. Acad. Sinica, 1983, 11, 309–328.

  5. Chen, B. Y. Total Mean Curvature and Submanifolds of Finite Type. World Scientific, Singapore, 1984.

  6. Chen, B. Y., Deprez, J., and Verheyen, P. Immersions with geodesics of 2-type. In Geometry and Topology of Submanifolds, IV, Proceedings of the Conference on Differential Geometry and Vision, Leuven 27–29 June 1991 (Dillen, F., ed.). World Scientific, Singapore, 1992, 87–110.

  7. Deprez, J., Dillen, F., and Verstraelen, L. Finite type space curves. Soochow J. Math., 1986, 12, 1–10.

  8. Gluck, H. Higher curvatures of curves in Euclidean space. Am. Math. Monthly, 1966, 73, 699–704.
doi:10.2307/2313974

  9. Hayden, H. A. On a generalized helix in a Riemannian n-space. Proc. London Math. Soc., 1931, 32, 37–45.
doi:10.1112/plms/s2-32.1.337

10. Klein, F. and Lie, S. Über diejenigen ebenen Curven welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergeben. Math. Ann., 1871, 4, 50–84.
doi:10.1007/BF01443297

11. Monterde, J. Curves with constant curvature ratios. 2007, 13, arXiv:math/0412323v1.

12. Özdamar, E. and Hacısalihoğlu, H. H. PA characterization of inclined curves in Euclidean n-space. Comm. Fac. Sci-Univ. Ankara, Ser. Al. Math., 1975, 24, 15–23.

13. Rodrigues Costa, S. On closed twisted curves. Proc. Am. Math. Soc., 1990, 109, 205–214.
doi:10.2307/2048380

14. Romero-Fuster, M. C. and Sanabria-Codesal, E. Generalized helices, twistings and flattenings of curves in n-space. Mat. Contemporanea, 1999, 17, 267–280.

15. Uribe-Vargas, R. On singularites, “perestroikas” and differential geometry of space curve. Ens. Math., 2004, 50, 69–101.

16. Weisstein, E. W. Ordinary Differential Equation – System with Constant Coefficients. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Ordinary Differential Equation System with Constant Coefficients.html (accessed 15 Sept. 2008).


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