Then, the concept of the h-hyperelastic Riemannian map is defined and using these findings, the Riemannian map along horizontal hyperelastic curves is characterized.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
💡
Sonuç & Katkı:Then, the concept of the h-hyperelastic Riemannian map is defined and using these findings, the Riemannian map along horizontal hyperelastic curves is characterized.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Alan Parametreleri
Abstract
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.
🔖
Bu Makaleye Atıf Yap
TÜKEL, G. Ö., TURHAN, T., ŞAHİN, B. (2022). Hyperelastic curves along Riemannian maps. Turkish Journal of Mathematics, Cilt: 46 - Sayı: 4, 1256–1267. https://ajindex.com/articles/607287
Arroyo J, Garay OJ, Barros M. Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces. Israel Journal of Mathematics 2003
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
Fischer AE. Riemannian maps between Riemannian manifolds. Contemporary Mathematics 1992
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
Ikawa T. On some curves in Riemannian geometry. Soochow Journal of Mathematics 1981
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