Riemannian and Finslerian spheres with fractal cut loci

TL;DR

This paper demonstrates the construction of smooth Riemannian and Finslerian metrics on spheres where the cut locus of a point can be a fractal, revealing complex geometric structures.

Contribution

It introduces a method to create highly differentiable metrics on spheres with fractal cut loci, extending to both Riemannian and Finsler geometries.

Findings

Existence of k-differentiable Riemannian metrics with fractal cut loci

Extension of the construction to Finsler spheres

Fractal structures in geometric cut loci

Abstract

The present paper shows that for a given integer k greater than 2 it is possible to construct an at least k-differentiable Riemannian metric on the sphere of a certain dimension such that the cut locus of a point of it becomes a fractal. Moreover, we show that this construction can be extended to the case of Finsler sphere as well.