Curvature bound and trajectories for magnetic fields on a Hadamard surfacs

Introduction.On a complete oriented Riemannian manifold M , a closed 2-form B is called a magnetic field.Let Ω denote the skew symmetric operator Ω:TM→ TM defined by u,Ω(v)=B(u, v) for every $u,$ v∈ TM .We call a smooth curve γ a trajectory for B if it satisfies the equation ∇γγ̇=Ω(γ̇) .Since Ω is skew symmetric, we find that every trajectory has constant speed and is defined for -∞<t<∞ .We shall call a trajectory normal if it is parametrized by its arc length.When γ is a trajectory for B , the curve σ defined by σ(t)=γ(λ t) with