The theorem of E. Hopf under uniform magnetic fields

摘要

By using a geodesic flow, E. Hopf proved that the total curvature of a compact surface without conjugate points is nonpositive, and vanishes if and only if the surface is flat in [5]. L. W. Green extended the result of E. Hopf for a compact n-dimensional manifold in [3]. Recently, F. Guimaraes and N. Innami have treated the noncompact case. The non-existence of a pair of conjugate points along geodesics is equivalent to the non-existence of singular values of the exponential map. If there exists a magnetic field, then this equivalence no longer holds. From this fact, we find two concepts of non-conjugation for the magnetic flow, which are called Jacobi field non-conjugation and exponential map non-conjugation.