Geodesics and magnetic curves in the 4-dim almost Kähler model space F4

Abstract

Abstract We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .