Projection onto the indicatrix bundle of a Finsler manifold

Abstract

We present a simple geometric description of the process "projection onto the indicatrix bundle" of a Finsler manifold originally introduced by H. Izumi, in the framework of the theory whose foundations were laid by J. Grifone [7], [8]. Next we derive an intrinsic expression for the projection of the Douglas tensor and give a new proof of Sakaguchi's important observation that the projected Douglas tensor and the Douglas tensor vanish together. Finally we deduce a concise coordinate-free form for the mixed curvature of the "Landsberg-Douglas manifolds", and conlude that in n > 2 dimensions these manifolds are just the Berwald manifolds.