A metrical approach to Finsler geometry

In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the non-linear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metric foundation to Finsler geometry and hence justifies the claim that ``Finsler geometry is Riemannian geometry without the quadratic restriction. The paper also contains a study of the compatibility condition to be placed between the metric and the non-linear connection.