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.