I'm working on something where I'm using the notion of Landsberg angle in Finsler geometry. I will expose some prerequisites in order to make my question more clear.
A Finsler structure in a manifold $M$ is just a nonnegative function $F:TM\longrightarrow\mathbb{R}$ wich is positively homogeneous in $v$ and such that each $g_p(v)=F^2(p,v)$ has positive definite Hessian matrix for all $v\neq 0$.
http://en.wikipedia.org/wiki/Finsler_manifold
It allows us to endow $TM-\{0\}$ with a Riemannian metric $g$. The set $S_p:=F^{-1}(1)\cap T_pM$ will be called the indicatrix of $F$ in $p$ and is a convex hypersurface of $T_pM$, diffeomorphic to an (n-1)-sphere. If we restrict the metric $g$ to $S_p$, so $(S_p,g)$ is a Riemannian manifold, whose distance I will call $d$. Then for any pair of nonzero vectors $v,w\in S_p$, we can define a certain ''angle'' doing $$\sphericalangle_p(v,w)=d(v,w).$$
We say that a diffeomorphism $\phi:M\longrightarrow M$ is an isometry if $F(p,v)=F\big(\phi(p),d\phi_p(v)\big)$. My question is, then, a very natural one: is the Landsberg angle invariant under isometries?
Any reference in the subject will be really appreciated! Thanks a lot for any help!