Principal normal spectral variations of space curves
DOI:
https://doi.org/10.22199/S0716-09172000000200004Keywords:
Spectral variational theory, vector fields, torsion, curvature, teoría espectral variacional, campos vectoriales, torsión, curvatura.Abstract
In the present paper, we give a similar spectral variational theory for closed curves in the Euclidean 3–space E3, considering deformations in the direction of the principal normal vector field. Similarly as in the planar case, the closed Euclidean space curves satisfying the corresponding variational minimal principle are characterized by their curvature being a function of finite Chen type; their torsion remains completely free.References
[1] D. E. Blair, Classification of 3-type curves. Soochow J.Math., 21, pp. 145–158, (1995).
[2] B. Y. Chen, Total mean curvature and submanifolds of finite type. World Scientific. Singapore, (1984).
[3] B. Y.Chen, J.Deprez, F.Dillen, L.Verstraelen and L.Vrancken, Curves of finite type. Geometry and Topology of Submanifolds, II, World Scientific. Singapore, pp. 76–110, (1990).
[4] B. Y. Chen, F.Dillen and L.Verstraelen, Finite type space curves. Soochow J.Math., 12, pp. 1–10, (1986).
[5] B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, A variational minimal principle characterizes submanifolds of finite type CR Acad. Sc. Paris 317, pp. 961–965, (1993).
[6] B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Compact hypersurfaces determined by a spectral variational principle Kyushu Journal of Math. 49, pp. 103–121, (1995).
[7] J. Deprez, F. Dillen and L. Vrancken, Finite type curves on quadrics Chinese J. Math., 18, pp. 95–121, (1990).
[8] M. Petrovic, L. Verstraelen and L. Vrancken, 3–type curves on ellipsiods of revolution Preprint series, Dept. Math. Katholieke Univ. Leuven, 2, pp. 31–49, (1990).
[9] M. Petrovic, L. Verstraelen and L. Vrancken, 3–type curves on hyperboloids of revolution and on cones of revolution Publ. Inst. Math. Belgrade, 59 (73), pp. 138–152, (1996).
[10] P. D. Scofield, Curves of constant precession Amer. Math. Monthly 102 (6), pp. 531–537, (1995).
[2] B. Y. Chen, Total mean curvature and submanifolds of finite type. World Scientific. Singapore, (1984).
[3] B. Y.Chen, J.Deprez, F.Dillen, L.Verstraelen and L.Vrancken, Curves of finite type. Geometry and Topology of Submanifolds, II, World Scientific. Singapore, pp. 76–110, (1990).
[4] B. Y. Chen, F.Dillen and L.Verstraelen, Finite type space curves. Soochow J.Math., 12, pp. 1–10, (1986).
[5] B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, A variational minimal principle characterizes submanifolds of finite type CR Acad. Sc. Paris 317, pp. 961–965, (1993).
[6] B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Compact hypersurfaces determined by a spectral variational principle Kyushu Journal of Math. 49, pp. 103–121, (1995).
[7] J. Deprez, F. Dillen and L. Vrancken, Finite type curves on quadrics Chinese J. Math., 18, pp. 95–121, (1990).
[8] M. Petrovic, L. Verstraelen and L. Vrancken, 3–type curves on ellipsiods of revolution Preprint series, Dept. Math. Katholieke Univ. Leuven, 2, pp. 31–49, (1990).
[9] M. Petrovic, L. Verstraelen and L. Vrancken, 3–type curves on hyperboloids of revolution and on cones of revolution Publ. Inst. Math. Belgrade, 59 (73), pp. 138–152, (1996).
[10] P. D. Scofield, Curves of constant precession Amer. Math. Monthly 102 (6), pp. 531–537, (1995).
Published
2017-06-14
How to Cite
[1]
M. Petrovic, J. Verstraelen, and L. Verstraelen, “Principal normal spectral variations of space curves”, Proyecciones (Antofagasta, On line), vol. 19, no. 2, pp. 141-155, Jun. 2017.
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