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On a complete oriented Riemannian manifold M, a closed 2-form B is called a magnetic field. Let Q denote the skew symmetric operator Q,: TM ―>TM defined by (u,Q(v)) = B(u, v) for every u, veTM. We call a smooth curve7 a trajectory for J? if it satisfies the equation V^7 = Q(y). Since O is skew symmetric, we find that every trajectory has constant speed and is defined for ―00< t< 00. We shall call a trajectory normal if it is parametrized by its arc length. When y is a trajectory for B, the curve G defined by R(t) = y(Xt) with some constant A is a trajectory for AB. We call the norm \\BX\\ of the operator Bx:TxMxTxM ―>R the strength of the magnetic field at the point x. For the trivialmagnetic field B = 0, the case without the force of a magnetic field, trajectories are nothing but geodesies. In term of physics it is a trajectory of a charged particleunder the action of the magnetic field.For a classical treatment and physical meaning of magnetic fieldssee [8]. On a Riemann surface M we can write down B = f -Vo\M with a smooth function/and the volum form VolM on M. When/is a constant function, the case

Curvature bound and trajectories for magnetic fields on a Hadamard surfacs