Hopf conjecture holds for analytic, k -basic Finsler tori without conjugate points

TLDR: In this paper, it was shown that analytic, k-basic Finsler metrics in the two torus without conjugate points are analytically integrable, in the sense that the unit tangent bundle of the metric admits an analytic foliation by invariant Lagrangian graphs.

Abstract: We show that analytic, k-basic Finsler metrics in the two torus without conjugate points are analytically integrable, in the sense that the unit tangent bundle of the metric admits an analytic foliation by invariant Lagrangian graphs. This result, combined with the fact that C1,L integrable k-basic Finsler metrics in the two torus have zero flag curvature (Barbosa-Ruggiero [19]) implies that analytic k-basic Finsler metrics in two tori without conjugate points are flat, a positive answer to the so-called Hopf conjecture for tori without conjugate points. Since there are well known examples of non flat tori without conjugate points (Busemann was the first to show such examples) the Hopf conjecture is not true if we drop the k-basic assumption. As for higher dimensional tori, a quite simple argument based on Schur’s Lemma shows that the only Finsler, k-basic (3 + m)-tori are the flat ones for every m ≥ 0.