Is a differentiable manifold with a metric a Finsler space?

Question

Let $M$ be a differentiable manifold and $d$ a metric on $M$ such that $d:M\times M\rightarrow \mathbb{R}$ is $C^\infty$. Is there some way $d$ will induce on $M$ a Finsler norm?

Answer 1

Suppose that $(M,f)$ is a smooth manifold with Finsler structure $f$, i.e. a (possibly nonsymmetric) norm defined on tangent spaces $T_pM, p\in M$ and which depends smoothly on $p$. Given $f$ one defines the "Finsler distance function" $d_f: M\times M\to R_+$ by $$ d_f(p,q)=\inf_{c} \int_0^1 f(c'(t))dt $$ where the infimum is taken over all smooth paths $c: [0,1]\to M$ connecting $p$ to $q$. If I understand your question correctly, you are asking which distance functions $d$ on $M$ appear as Finsler distance functions. Necessary conditions for this are that $d$ is a path-metric, existence of geodesics between nearby points, existence of local prolongation of geodesics.

A partial answer (the above conditions plus a bit more, like uniqueness of prolongation of geodesics, which means that we are dealing with smooth norms on $P_pM$) are sufficient for the existence of a Finsler metric $f$ such that $d=d_f$) was given by Busemann in

H. Busemann, "Recent synthetic differential geometry." Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 54. Springer-Verlag, New York-Berlin, 1970. vii+110 pp.

I am not an expert in Finsler geometry, so my understanding of his results is limited. But, in the non-Riemannian setting, I think, Busemann's work is the best known result. In the Riemannian setting, the definitive result is due to Nikolaev:

I.G. Nikolaev, Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Sibirsk. Mat. Zh. 24 (1983), no. 2, 114–132.