Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves (Q818782)

A smooth curve \(\gamma= \gamma(s)\) in a complex \(n\)-dimensional Kähler manifold \((M,J)\) parametrized by its arc length \(s\) is called a Kähler Frenet curve, if it satisfies \(\nabla_{\dot\gamma}\dot\gamma= \kappa(s) J\dot\gamma\) or \(\nabla_{\dot\gamma}\dot\gamma= -\kappa(s)J\dot\gamma\). Let \(f\) be a Kähler isometric immersion of a Kähler manifold \(M_n\) into an arbitrary Kähler manifold \(\widetilde M_m\) (resp. into a real space form \(\widetilde M^{2n+p}(\widetilde c)\)). Assume that there exists a positive \(C^\infty\)-function \(\kappa= \kappa(s)\) such that \(f\) maps every Kähler Frenet curve \(\gamma= \gamma(s)\) of curvature \(\kappa\) on \(M_n\) to a Frenet curve of order 2 in \(\widetilde M_m\) (resp. by non-constant \(\kappa\) to a plane curve). It is proved that then \(f\) is a totally geodesic immersion. Finally, every parallel isometric immersion of a complex space form \(M_n(c)\) into a real space form \(\widetilde M^{2n+p}\) is characterized in terms of Kähler Frenet curves.