Harmonic vector fields on Landsberg manifolds

Abstract

Let (M,F) be a compact boundaryless Landsberg manifold. In this work, a necessary and sufficient condition for a vector field on (M,F) to be harmonic is obtained. Next, on a compact boundaryless Finsler manifold of zero flag curvature, a necessary and sufficient condition for a vector field to be harmonic is found. Furthermore, the nonexistence of harmonic vector fields on a compact Landsberg manifold is studied and an upper bound for the first de Rham cohomology group is obtained.