Distance from a Finsler Submanifold to its Cut Locus and the Existence of a Tubular Neighborhood

TL;DR

This paper proves the positivity of the cut time map for submanifolds in Finsler manifolds, establishing the existence of tubular neighborhoods and providing new geometric proofs that simplify previous results.

Contribution

It offers a novel geometric proof for the positivity of the cut time map and the existence of tubular neighborhoods in Finsler manifolds, improving upon prior work.

Findings

Cut time map is always positive for submanifolds in Finsler manifolds.

Existence of a tubular neighborhood around submanifolds is established.

Simplified proofs of previous results under weaker assumptions.

Abstract

In this article we prove that for a closed, not necessarily compact, submanifold $N$ of a possibly non-complete Finsler manifold $(M, F)$, the cut time map is always positive. As a consequence, we prove the existence of a tubular neighborhood of such a submanifold. When $N$ is compact, it then follows that there exists an $\epsilon > 0$ such that the distance between $N$ and its cut locus $\mathrm{Cu}(N)$ is at least $\epsilon$. This was originally proved by B. Alves and M. A. Javaloyes (Proc. Amer. Math. Soc. 2019). We have given an alternative, rather geometric proof of the same, which is novel even in the Riemannian setup. We also obtain easier proofs of some results from N. Innami et al. (Trans. Amer. Math. Soc., 2019), under weaker hypothesis.