Let $M$ be a Riemannian manifold and let $N$ be a submanifold of $M$. Then the cut locus of $N$, denoted by $\mathrm{Cu}(N)$, is the set of all points beyond which a distance minimal geodesic fails to be distance minimal. We know that $\mathrm{Cu}(N)\cap N=\emptyset$.
I want to know whether the same is true if $M$ is a Finsler manifold. Also, like in Riemannian manifold, a distance minimal geodesic is perpendicular to the submanifold. Is there any similar condition for the Finsler manifold?