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

Aritra Bhowmick, Sachchidanand Prasad

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 $ε> 0$ such that the distance between $N$ and its cut locus $\mathrm{Cu}(N)$ is at least $ε$. 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.

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

Abstract

In this article we prove that for a closed, not necessarily compact, submanifold of a possibly non-complete Finsler manifold , the cut time map is always positive. As a consequence, we prove the existence of a tubular neighborhood of such a submanifold. When is compact, it then follows that there exists an such that the distance between and its cut locus is at least . 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.

Paper Structure

This paper contains 23 sections, 19 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Finsler Geometry
  3. Finsler Metric
  4. Legendre Transformation
  5. Chern Connection
  6. Geodesics
  7. Submanifolds in Finsler Manifolds
  8. Principal Curvature
  9. N-Geodesics
  10. Cut Locus of a Submanifold
  11. Separating Points and Focal Points of a Submanifold
  12. Principal Curvatures of Small Backward Spheres
  13. Geometric Tubular Neighborhood of a Submanifold
  14. Finslerian Hessian
  15. Minimizing the Distance Function to a Point from a Submanifold
  16. ...and 8 more sections

Key Result

Theorem A

Given a closed (not necessarily compact) submanifold $N$ of a Finsler manifold $(M, F)$, we have $N \cap \mathrm{Cu}(N) = \emptyset$.

Figures (4)

  • Figure 1: A sufficiently small sphere intersects the submanifold at a unique point.
  • Figure 2: In local coordinates near $p$, the hypersurfaces $S$ and $P$ are graphs of $h^S$ and $h^P$.
  • Figure 3: For the point $q^\prime$ on $\gamma_{\mathbf{n}}$, the distance from $P$ is achieved at $x^\prime$.
  • Figure 4: A counterexample : we always have $q$ arbitrarily near to $p$ for which the distance to $N$ is achieved at two distinct points.

Theorems & Definitions (50)

  • Theorem A: \ref{['cor:cutLocusDisjoint']}
  • Theorem B: \ref{['thm:tubularNBD']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • ...and 40 more