Relaxed elastic line in a Riemannian manifold

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

We obtain a differential equation with 2 boundary conditions for a relaxed elastic line in a Riemannian manifold. This differential equation, which is found with respect to constant sectional curvature G, geodesic curvature k, and 2 boundary conditions, gives a more direct and more geometric approach to questions concerning a relaxed elastic line in a Riemannian manifold. We give various theorems and results in terms of a relaxed elastic line. Consequently, we examine the concept of a relaxed elastic line in 2- and 3- dimensional space forms.

Anahtar Kelimeler:

Relaxed elastic line, Riemannian manifold, geodesic curvature, space forms

Relaxed elastic line in a Riemannian manifold

Keywords:

Relaxed elastic line, Riemannian manifold, geodesic curvature, space forms,

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do Carmo, M.P.: Riemannian Geometry. Boston. Birkh¨ auser 1992 .
Jurdjevic, V.: Geometric Control Theory. New York. Cambridge University Press 1997 .
K¨ uhnel, W.: Differential Geometry: Curves - Surfaces - Manifolds. Providence. American Mathematical Society 2006 .
Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Diff. Geom. 1984; 20, 1–22.
Manning, G.S.: Relaxed elastic line on a curved surface. Q. Appl. Math. 1987; 45, 515–527.
Nickerson, H.K., Manning, G.S.: Intrinsic equations for a relaxed elastic line on an oriented surface. Geometriae Dedicata 1988; 27, 127–136.
Singer, D.: Lectures on elastic curves and rods. AIP Conf. Proc. 1002. Melville, New York. Amer. Inst. Phys. 2008. Weinstock, R.: Calculus of Variations. New York. Dover Publications 1974 .