A Condition for Classical Elastic Curves on Surface

Year 2018, Volume: 6 Issue: 1, 43 - 49, 27.04.2018

Ayşe Altın

https://doi.org/10.36753/mathenot.421755

Abstract

In this paper, we consider two fixed points p to q on a Riemannian surface M in 3-dimensional Euclidean
space. We obtain a condition for classical elastic curves with in the family of all curves from p to q on M.
We also prove that this condition can be expressed in terms of the curvature functions. The condition is
realized for curves whose geodesic and normal curvature functions are both constant. 

Keywords

Energy, Energy of a unit vector field, Elastica

References

• [1] M. Barros, A. Ferrndez, P. Lucas and M. A. Meroo, "Willmore Tori and Willmore-Chen Submanifolds in Pseudo-Riemannian Spaces", Journal of Geometry and Physics, Vol. 28, pp. 45-66, 1998.
• [2] D. H. Steinberg, "Elastic Curves in Hyperbolic Space", Doctoral Thesis, Case Western Reserve University, UMI Microform 9607925, 72p, 1995.
• [3] R. Huang, "A Note on the p-elastica in a Constant Sectional Curvature Manifold", Journal of Geometry and Physics, Vol. 49, pp. 343-349, 2004.
• [4] A. E. H. Love„ "A treatise on the mathematical theory of elasticity", Cambridge University Press, 2013.
• [5] H. Gluck, W. Ziller, "On the volume of the unit vector fields on the three sphere", Comment Math. Helv, Vol. 61, pp. 177-192, 1986.
• [6] D. L. Johnson, "Volume of flows", Proc. Amer. Math. Soc, Vol. 104, pp. 923-932, 1988.
• [7] A. Higuchi, B. S. Kay, C. M. Wood, "The energy of unit vector fields on the 3-sphere". Journal of Geometry and Physics, Vol. 37, pp. 137-155, 2001.
• [8] C. M. Wood, "On the Energy of a Unit Vector Field", Geometrae Dedicata, Vol. 64, pp. 319-330 1997.
• [9] B. O’Neill, "Elementary Diffrential Geometry", Academic Press Inc., 1966.
• [10] D. J. Struik, "Differential Geometry", Reading MA: Addison-Wesley, 1961.
• [11] A. Altın, "On the Energy of Frenet Vectors Fields in Rn", Cogent Mathematics, 2017.
• [12] P. A. Chacón, A. M. Naveira, J. M. Weston, "On the Energy of Distributions, with Application to the Quarternionic Hopf Fibration", Monatshefte fûr Mathematik, Vol. 133, pp. 281-294, 2001.
• [13] T. Sakai, "Riemannian Geometry", American Mathematical Society, 1996.
• [14] P. M. Chacón, A. M. Naveira, "Corrected Energy of Distributions on Riemannian Manifold", Osaka Journal Mathematics, Vol 41, pp. 97-105, 2004.
• [15] L. Euler, "Additamentum de curvis elasticis. In Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gaudentes". Lausanne, 1744.
• [16] P. Baird, J. C. Wood, "Harmonic Morphisms Between Riemannian Manifold", Clarendos press, Oxford, 2003.