We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution.
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References
P. M. Chacón and A. M. Naveira, “Corrected energy of distributions on Riemannian manifold,” Osaka J. Math., 41, 97–105 (2004).
A. Higuchi, B. S. Kay, and C. M. Wood, “The energy of unit vector fields on the 3-sphere,” J. Geom. Phys., 37, 137–155 (2001).
C. M. Wood, “On the energy of a unit vector field,” Geom. Dedic., 64, 319–330 (1997).
A. Altın, “On the energy and pseudo-angle of Frenet vector fields in R n υ ,” Ukr. Math. J., 63, No. 6, 833–839 (2011).
P. M. Chacón, A. M. Naveira, and J. M.Weston, “On the energy of distributions, with application to the quaternionic Hopf fibration,” Monatsh. Math., 133, 281–294 (2001).
B. O’Neill, Elementary Differential Geometry, Academic Press, New York (1966).
D. J. Struik, Differential Geometry, Addison-Wesley, Reading, MA (1961).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 4, pp. 568–573, April, 2015.
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Altın, A. The Energy of a Domain on the Surface. Ukr Math J 67, 641–647 (2015). https://doi.org/10.1007/s11253-015-1128-7
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DOI: https://doi.org/10.1007/s11253-015-1128-7