Study of bi-f-harmonic curves and helices along Riemannian map

Abstract

In this paper, we study bi-f-harmonic curves and helices along the Riemannian map. We find that, if a totally umbilical Riemannian map takes a horizontal bi-f-harmonic curve to bi-f-harmonic curve, then the map is totally geodesic.
Then, we discuss the mean curvature vector field for horizontal bi-harmonic curves through Riemannian maps. In addition, we obtain the condition for the curvature of helix along isotropic Riemannian map.