TRANSITIVITY OF FINSLER GEODESIC FLOWS OF COMPACT SURFACES WITHOUT CONJUGATE POINTS AND HIGHER GENUS, AND APPLICATIONS TO FINSLER RIGIDITY PROBLEMS

摘要

We show that if (M, F) is a C-infinity k-basic Finsler compact surface without conjugate points and genus greater than one whose Green bundles are continuous then (M, F) is Riemannian. The proof combines Riemann-Finsler geometry, the theory of Lagrangian bundles and the transitivity of Finsler geodesic flows of compact surfaces of genus greater than one. The transitivity of geodesic flows of compact Finsler surfaces without conjugate points and higher genus generalizes the work of P. Eberlein for compact Riemannian surfaces without conjugate points in the context of visibility manifolds.