The cut locus and distance function from a closed subset of a Finsler manifold

2012/07/04 by Minoru Tanaka, Tanaka, Minoru, Sorin V. Sabău +2 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #math.DG

paper · pdf · doi:10.48550/arxiv.1207.0918

openalex publication_date 2012/07/04 · arxiv created 2012/12/15 · arxiv updated 2012/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We characterize the differentiable points of the distance function from a closed subset N of an arbitrary dimensional Finsler manifold in terms of the number of N-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset N, namely that it is a local tree, it is made of countably many rectifiable Jordan arcs except for the endpoints of the cut locus and that an intrinsic metric can be introduced in the cut locus and its intrinsic and induced topologies coincide. We should point out that these are new results even for Riemannian manifolds.

Cited by

• The cut locus of a Randers rotational 2-sphere of revolution
• On Closed Geodesics in Non-negatively Curved Finsler Structures with Large Volume Growth

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