Research Article
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Year 2018, Volume: 6 Issue: 1, 43 - 49, 27.04.2018
https://doi.org/10.36753/mathenot.421755

Abstract

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.

A Condition for Classical Elastic Curves on Surface

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

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.
There are 16 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Ayşe Altın

Publication Date April 27, 2018
Submission Date November 28, 2017
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA Altın, A. (2018). A Condition for Classical Elastic Curves on Surface. Mathematical Sciences and Applications E-Notes, 6(1), 43-49. https://doi.org/10.36753/mathenot.421755
AMA Altın A. A Condition for Classical Elastic Curves on Surface. Math. Sci. Appl. E-Notes. April 2018;6(1):43-49. doi:10.36753/mathenot.421755
Chicago Altın, Ayşe. “A Condition for Classical Elastic Curves on Surface”. Mathematical Sciences and Applications E-Notes 6, no. 1 (April 2018): 43-49. https://doi.org/10.36753/mathenot.421755.
EndNote Altın A (April 1, 2018) A Condition for Classical Elastic Curves on Surface. Mathematical Sciences and Applications E-Notes 6 1 43–49.
IEEE A. Altın, “A Condition for Classical Elastic Curves on Surface”, Math. Sci. Appl. E-Notes, vol. 6, no. 1, pp. 43–49, 2018, doi: 10.36753/mathenot.421755.
ISNAD Altın, Ayşe. “A Condition for Classical Elastic Curves on Surface”. Mathematical Sciences and Applications E-Notes 6/1 (April 2018), 43-49. https://doi.org/10.36753/mathenot.421755.
JAMA Altın A. A Condition for Classical Elastic Curves on Surface. Math. Sci. Appl. E-Notes. 2018;6:43–49.
MLA Altın, Ayşe. “A Condition for Classical Elastic Curves on Surface”. Mathematical Sciences and Applications E-Notes, vol. 6, no. 1, 2018, pp. 43-49, doi:10.36753/mathenot.421755.
Vancouver Altın A. A Condition for Classical Elastic Curves on Surface. Math. Sci. Appl. E-Notes. 2018;6(1):43-9.

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