A note on the existence of tubular neighbourhoods on Finsler manifolds and minimization of orthogonal geodesics to a submanifold

TL;DR

This paper establishes the existence of tubular neighborhoods and smooth distance functions around submanifolds in Finsler manifolds, along with properties of orthogonal geodesics minimizing distance.

Contribution

It proves the existence of tubular neighborhoods and smooth distance functions in Finsler manifolds, extending classical Riemannian results to the Finsler setting.

Findings

Orthogonal geodesics minimize distance locally.

Existence of tubular neighborhoods around submanifolds.

Distance from the submanifold is smooth in a neighborhood.

Abstract

In this note, we prove that given a submanifold $P$ in a Finsler manifold $(M,F)$, (i) the orthogonal geodesics to $P$ minimize the distance from $P$ at least in some interval, (ii) there exist tubular neighbourhoods around each point of $P$, (iii) the distance from $P$ is smooth in some open neighbourhood of $P$ (but not necessarily in $P$).