Gözde ÖZKAN TÜKEL, Tunahan TURHAN, Bayram ŞAHİN

Hyperelastic curves along Riemannian maps

The main purpose of this paper is to examine what kind of information the smooth Riemannian map defined between two Riemannian manifolds provides about the character of the Riemannian map when a horizontal hyperelastic curve on the total manifold is carried to a hyperelastic curve on the base manifold. For the solution of the mentioned problem, firstly, the behavior of an arbitrary horizontal curve on the total manifold under a Riemannian map is investigated and the equations related to pullback connection are obtained. The necessary conditions are given for the Riemannian map to be h-isotropic or totally umbilical when a horizontal Frenet curve in the total manifold transforms to a hyperelastic curve on the base manifold. Then, the concept of the h-hyperelastic Riemannian map is defined and using these findings, the Riemannian map along horizontal hyperelastic curves is characterized.

PDF

___

[1] Arroyo J, Garay OJ, Barros M. Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces. Israel Journal of Mathematics 2003; 138: 171-187. doi: 10.1007/BF02783425
[2] Arroyo J, Garay OJ, Mencia JJ. Closed free hyperelastic curves in real space forms. Proceeding of the XII Fall Workshop on Geometry and Physics, Coimbra, 2003; 1-13.
[3] Fischer AE. Riemannian maps between Riemannian manifolds. Contemporary Mathematics 1992; 132: 331-366. doi:10.1090/conm/132/1188447
[4] Garay OJ. Riemannian submanifolds shaped by the bending energy and its allies. Proceedings of The Sixteenth International Workshop on Differential Geometry 2012; 16: 57-70.
[5] Ikawa T. On some curves in Riemannian geometry. Soochow Journal of Mathematics 1981; 7: 37-44.
[6] Langer J, Singer DA. The total squared curvature of closed curves. Journal of Differential Geometry 1984; 20 (1): 1-22. doi:10.4310/jdg/1214438990
[7] Nomizu K, Yano K. On circles and spheres in Riemannian geometry. Mathematische Annalen 1974; 210 (2): 163-170.
[8] O’Neill B. The fundamental equations of a submersion. Michigan Mathematical Journal 1966; 13 (4): 459-469. doi:10.1307/mmj/1028999604
[9] Özkan Tükel G, Turhan T, Yücesan A. Hyperelastic Lie quadratics. Honam Mathematical Journal 2019; 41 (2): 369-380. doi:10.5831/HMJ.2019.41.2.369
[10] Popiel T, Noakes L. Elastica in SO(3). Journal of Australian Mathematical Society 2007; 83: 105-124. doi:10.1017/S1446788700036417
[11] Singer DA. Lectures on elastic curves and rods. In AIP Conference Proceedings American Institute of Physics 2008; 41 (2): 369-380.
[12] Şahin B. Riemannian Submersions, Riemannian Maps in Hermitian Geometry and Their Applications. Elsevier, 2017.
[13] Şahin B, Özkan Tükel G, Turhan T. Hyperelastic curves along immersions. Miskolc Mathematical Notes 2021; 22 (2): 915-927. doi:10.18514/MMN.2021.3501
[14] Özkan Tükel G, Turhan T, Şahin B. Isotropic Riemannian maps and helices along Riemannian maps, arXiv 2105.10119, 2021.