1. Introduction

Finsler geometry is an essential extension of Riemannian geometry. Instead of an inner product on every tangent space one considers Minkowski norms on every tangent space. For a Finsler metric the unit sphere in each tangent space is a strictly convex hypersurface. One obtains for every nonzero tangent vector an inner product, arising from Minkowski norm; in the Riemannian case these inner products all coincide on a fixed tangent space. The length of a smooth curve is well-defined. Geodesics—locally length-minimizing curves parametrized with constant speed—are uniquely defined for a given initial direction. From the viewpoint of the calculus of variations Finsler metrics are a suitable generalization of Riemannian metrics such that the variational problem for the length of curves between two fixed points is positive and positive regular. In terms of physics a Finsler metric describes a Lagrangian system without a potential; a Riemannian metric can be viewed as the special case of quadratic kinetic energy.

But in contrast to the Riemannian case there is no canonical connection, so several connections have been used in Finsler geometry. We use here the one introduced by S.-S. Chern [Bao et al. 2000, Chapter 2], transposed to vector fields on the manifold for a fixed direction field: Given a nowhere vanishing vector field V in an open nonempty subset U, there is a uniquely determined torsionfree connection ∇V that is almost metric. Using this connection one can define the flag curvature, which generalizes the sectional curvature in Riemannian geometry and controls the infinitesimal behavior of geodesics.