Finslerian Diffusion and Curvature

Abstract

As in the well-known Riemannian case, one would expect Finslerian diffusion to be closely related to the curvature of the manifold. In the present chapter we establish such a relationship for Finslerian h-development. A major difficulty is that all methods of relating curvature and diffusion involve normal coordinates, which, unfortunately do not exist for all Finsler manifolds. This is because the exponential map from TM _x to M can, in general, develop a singularity at the origin. Because of this, we restrict ourselves to Berwald spaces, a class of Finsler spaces which can be characterized by the existence of normal coordinates at each point x ∈ M [Dou28].