Metric structures associated to Finsler metrics

Abstract

AbstractWe investigate the relation between weighted quasi-metric Spaces and FinslerSpaces. We show that the induced metric of a Randers space with reversiblegeodesics is a weighted quasi-metric space. 1 Introductionand Motivation Riemannian spaces can be represented as metric spaces. Indeed, for a Riemannian space(M,a) we can define the induced metric space (M,d α ), with the metricd α : M× M→ [0,∞), d α (x,y) := inf γ∈Γ xy Z ba α(γ(t),γ˙(t))dt, (1.1)where Γ xy := {γ: [a,b] → M| γ(piecewise) C ∞ -curve,γ(a) = x,γ(b) = y} is the set ofcurves joining points xand y, ˙γ(t) := dγ(t)dt the tangent vector to γat γ(t), and α(x,X)the Riemannian norm of the vector X∈ T x M. It is easy to see that d α is a metric on M,i.e. it satisfies the axioms:1. Positiveness: d α (x,y) >0 if x6= y, d α (x,x) = 0,2. Symmetry: d α (x,y) = d α (y,x),3. Triangle inequality: d α (x,y) ≤ d α (x,z) +d α (z,y),for any x,y,z∈ M.More general structures than Riemannian ones are Finsler structures (see [BCS00],[S01], [MHSS01] for definitions).Similarly with the Riemannian case, one can define the induced metric of a Finslerspace (M,F) byd